Addition

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

5

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

6

1

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

AdditionTo “add” means to combine two quantities A and B.

The digit–sum table (Wikipedia)

To add two numbers,

Example A. Add 8,978 + 657

8,978657+

1

53

1

6

1

9,So the sum is 9,635.

2. add the digits from right to left and “carry” when necessary.

The combined result is called the sum or the total of A and B.

All the following words mean to “add”: total, sum, combine, increase by, count up, aggregate, augmented by, tally, etc..

A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).

1. line up the numbers vertically to match the place values,

Addition

+

Addition

+

Addition

+ +

Addition

=+ +

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

In general, if A and B are two numbers, then A + B = B + A and we say that “the addition operation is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

+=

+

In general, if A and B are two numbers, then A + B = B + A and we say that “the addition operation is commutative.”

If we are to add two apples to a pile of three apples, the outcome is the same as adding three apples to the pile of two apples.

Addition

=

The subtraction operation is not commutative, that is,

– –≠

In practical terms, this means that when doing addition, we don’t care who is added to whom,

or A – B ≠ B – A

but when doing subtraction, be sure “who” is taken away from “whom.”

+ +

SubtractionTo subtract is to take away, or to undo an addition.

SubtractionTo subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A.

SubtractionTo subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A.We call the outcome “the difference of A and B” and we write A – B = D (for difference).

SubtractionTo subtract is to take away, or to undo an addition.

The following phrases are also translated as “A – B”: “A subtract B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away from A.”

We write “A – B” for taking the amount B away from A.We call the outcome “the difference of A and B” and we write A – B = D (for difference).

SubtractionTo subtract is to take away, or to undo an addition.

The following phrases are also translated as “A – B”: “A subtract B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away from A.”

We write “A – B” for taking the amount B away from A.

Hence the statements “five apples take away three apples,”

all mean 5 – 3

“three apples are taken away from five apples,”“five apples minus three apples,”

= 2 .

We call the outcome “the difference of A and B” and we write A – B = D (for difference).

SubtractionTo subtract is to take away, or to undo an addition.

If “who is taken away from whom” is not specified, then it is assumed that we are taking the smaller number away from the bigger one. So “the difference between $10 and $50” is 50 –10 = $40. (In fact, we can’t do 10 – 50, yet.)

The following phrases are also translated as “A – B”: “A subtract B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away from A.”

We write “A – B” for taking the amount B away from A.

Hence the statements “five apples take away three apples,”

all mean 5 – 3

“three apples are taken away from five apples”“five apples minus three apples,”

= 2 .

We call the outcome “the difference of A and B” and we write A – B = D (for difference).

SubtractionTo subtract, 1. lineup the numbers vertically,

SubtractionTo subtract, 1. lineup the numbers vertically,

For example, 634 – 87 is: 6 3 4 8 7–

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary.

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

125

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

125

45

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

125

45When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

125

45

Example A. The store price of a Thingamajig is $500. How much money do we save if we buy one for $400 online?

When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.

Subtraction

For example, 634 – 87 is: 6 3 4 8 7–

To subtract, 1. lineup the numbers vertically,2. subtract the digits from right to left and “borrow” when it is necessary. need to borrow

142

7

125

45

Example A. The store price of a Thingamajig is $500. How much money do we save if we buy one for $400 online? The amount saved is: the expensive price – the cheaper price, so we saved 500 – 400 = $100.

When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

108th floortop

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

108th floortop

1st hr 42th floor

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

108th floortop

1st hr 42th floor2nd hr 67th floor

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor. a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.

Nth fl.

108th fl.

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.

Nth fl.

108th fl.

?

SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor.

After the 1st hour, we still have 108 – 42 = 66 floors to the top.

a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?

108th floortop

1st hr 42th floor2nd hr 67th floor

During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.We are on the Nth floor out of total 108 floors, so the number of remaining floors to the topis “108 – N” as shown. (Not “N – 108”!)

Nth fl.

108th fl.

108 – N

We simplify the notation for adding the same quantity repeatedly.

Multiplication

We simplify the notation for adding the same quantity repeatedly.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In general, just as addition, Multiplication s commutative, i.e. A x B = B x A.

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

In general, just as addition, Multiplication s commutative, i.e. A x B = B x A.

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

In general, just as addition, Multiplication s commutative, i.e. A x B = B x A.

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

the result 6 is called the product(of 2 and 3).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

In general, just as addition, Multiplication s commutative, i.e. A x B = B x A.

Multiplication

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

the result 6 is called the product(of 2 and 3).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

(Note: 1 and 6 are also factors of 6 because 1 x 6 = 6 x 1 = 6.)

In general, just as addition, Multiplication s commutative, i.e. A x B = B x A.

Multiplication

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

Multiplication

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

Multiplication

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

* (1 * x = x * 1 = x) The product of 1 with any number x is x.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

Multiplication

* For the products with 9 as a factor, the sum of their digits is 9.

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

* (1 * x = x * 1 = x) The product of 1 with any number x is x.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

Multiplication

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81For example,

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

* (1 * x = x * 1 = x) The product of 1 with any number x is x.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

all have digit sum equal to 9,

Multiplication

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81For example,

i.e. 5 + 4 = 9,

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

* (1 * x = x * 1 = x) The product of 1 with any number x is x.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

all have digit sum equal to 9,

Multiplication

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81For example,

i.e. 5 + 4 = 9, 6 + 3 = 9

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

* (1 * x = x * 1 = x) The product of 1 with any number x is x.

The multiplication table shown here is to be memorized and below are some features and tricks that might help.

7 + 2 = 9, 8 + 1 = 9all have digit sum equal to 9,

Multiplication

* The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

Multiplication

6636 6742 6848 69547749 7856 7963

8864 8972 9981

* The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

Multiplication

6636 6742 6848 69547749 7856 7963

8864 8972 9981

* The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

6 x 7 = 42 (= 7 x 6)For example,

7 x 8 = 56 (= 8 x 7).

Multiplication

6636 6742 6848 69547749 7856 7963

8864 8972 9981

* The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

6 x 7 = 42 (= 7 x 6)For example,

The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare called even numbers.

7 x 8 = 56 (= 8 x 7).

Multiplication

6636 6742 6848 69547749 7856 7963

8864 8972 9981

* The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

6 x 7 = 42 (= 7 x 6)For example,

The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare called even numbers.The numbers 0(= 0*0), 1(= 1*1), 4(= 2*2), 9(= 3*3), 16(= 4*4),.., of the form x*x, down the diagonal, are called square numbers.

7 x 8 = 56 (= 8 x 7).

Multiplication

The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done.

Multiplication

The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

Multiplication

The Vertical Format

47

7x

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

7x

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

7x

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 ii. 7x7=49,record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 ii. 7x7=49, 49+2=51

add the previous carry to the product,

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 ii. 7x7=49,

1

record the 1,

5

carry the 5

49+2=51

add the previous carry to the product,record the unit-digit of this sum and carry the 10’s digit of this sum.

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

We use a vertical format to multiply larger numbers. The following demonstrates how this is done.We start with a two-digit number times a single digit number.

i. 4x7=28 ii. 7x7=49,

1

record the 1,

5

carry the 5

49+2=51

add the previous carry to the product,record the unit-digit of this sum and carry the 10’s digit of this sum.

record the unit-digit of the product, and carry the 10’s digit of the product.

To multiply a longer number against a single digit number, repeat step ii until all the digits are multiplied.

Multiplication

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

47

7x

9

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

9For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28

9For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49, 49+2=51

9For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

49+2=51

9For example,

Multiplication carry the 5

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

6

For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

6When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

6When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

Multiplication

6

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

1

record the 1

9

8

record the 8

carry the 6

66

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

4x6=24

1

record the 1

9

8

record the 8

carry the 6

66

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

4x6=24

1

record the 1

←record

9

8

record the 8

carry the 6

66

carry the 2

4

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

4x6=24 7x6=42,

1

record the 1

←record

42+2=44 9

8

record the 8

carry the 6

66

carry the 2

4

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44 9

8

record the 8

carry the 6

66

carry the 2

44

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44 9

9x6=54 54+4= 58

8

record the 8

carry the 6

66

carry the 2

44

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44 9

9x6=54 54+4= 58

8

record the 8

carry the 6

66

carry the 2

4485

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44 9

9x6=54 54+4= 58

8

record the 8

carry the 6

66

carry the 2

Finally, we obtain the answer by adding the two rows.

4485

Multiplication

x

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left.

47

78

carry the 4

4x6=24 7x6=42,

1

←record

42+2=44 9

9x6=54 54+4= 58

866

carry the 2

Finally, we obtain the answer by adding the two rows.

44

85

85 2 6 5

Multiplication

+

x

Division is the operation of dividing a given amount into a prescribed number of equal parts.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q

The total T is the dividend,

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q

The total T is the dividend,

The number of parts D is the divisor.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q

The total T is the dividend,

The number of parts D is the divisor.

Q is the quotient.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q says that “if T is divided into D equal parts, then each part is Q.”

The total T is the dividend,

The number of parts D is the divisor.

Q is the quotient.

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q says that “if T is divided into D equal parts, then each part is Q.”

The total T is the dividend,

The number of parts D is the divisor.

Q is the quotient.

If T ÷ D = Q then T = D x Q or that D and Q are factors of T,

Division

Division is the operation of dividing a given amount into a prescribed number of equal parts.For example, if three people share a dozen apples, then each person gets four apples and there is no leftovers.

In this case, we say that “12 divides evenly by 3”. We write this as “12 ÷ 3 = 4” which translates into“if 12 is divided into 3 equal parts, then each part is 4”.In general, the expression

T ÷ D = Q says that “if T is divided into D equal parts, then each part is Q.”

The total T is the dividend,

The number of parts D is the divisor.

Q is the quotient.

If T ÷ D = Q then T = D x Q or that D and Q are factors of T, e.g. 12 ÷ 3 = 4 so 12 = 3(4), so both 3 and 4 are factors of 12.

Division

The Vertical Format Division

We demonstrate the vertical long-division format below.The Vertical Format Division

We demonstrate the vertical long-division format below.The Vertical Format

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

multiply the quotientback into the scaffold.

63 x 2

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

multiply the quotientback into the scaffold.

63 x 2 0

The new dividend is 0,

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder R. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

63 x 2 0

The new dividend is 0,

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder R. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

63 x 2 0

The new dividend is 0, not enough to be divided again, stop. This is the remainder R.

Division

We demonstrate the vertical long-division format below.The Vertical Format

Example C. a. Write 6 ÷ 2 as

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend.)

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.“back-one”

outside )2 6

“front-one” inside

Enter the quotient on top

3

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder R. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

63 x 2 0

The new dividend is 0, not enough to be divided again, stop. This is the remainder R.

So the remainder R is 0 and we have that 6 ÷ 2 = 3 evenly.

Division

b. Carry out the long division 7 ÷ 3.Division

b. Carry out the long division 7 ÷ 3.Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

Division

b. Carry out the long division 7 ÷ 3.Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Division

b. Carry out the long division 7 ÷ 3.Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

Division

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Division

Enter the quotient on top

2

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

multiply the quotientback into the scaffold.

62 x 3 1

Division

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

62 x 3 1

Division

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

62 x 3 1

The new dividend is 1, not enough to be divided again, so stop. This is the remainder.

Division

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

62 x 3 1

The new dividend is 1, not enough to be divided again, so stop. This is the remainder.

So the remainder is 1 and we have that 7 ÷ 3 = 2 with R = 1.

Division

b. Carry out the long division 7 ÷ 3.

ii. Enter the quotient on top,Multiply the quotient back into the problem and subtract the results from the dividend (and bring down the rest of the digits, if any. This is the new dividend).

Steps. i. (Front-in Back-out)Put the problem in the long division format with the back-number (the divisor) outside, and the front-number (the dividend) inside the scaffold.

“back-one” outside )3 7

“front-one” inside

Enter the quotient on top

2

iii. If the new dividend is not enough to be divided by the divisor, stop. This is the remainder. Otherwise, repeat steps i and ii.

multiply the quotientback into the scaffold.

62 x 3 1

The new dividend is 1, not enough to be divided again, so stop. This is the remainder.

So the remainder is 1 and we have that 7 ÷ 3 = 2 with R = 1. Put the result in the multiplicative form, we have that 7 = 2 x 3 + 1.

Division

Division

)3 7 7 4 3 1 7

c. Divide 74317 ÷ 37. Find the Q and R.

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice. 2

c. Divide 74317 ÷ 37. Find the Q and R.

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

2

c. Divide 74317 ÷ 37. Find the Q and R.

7 4

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

2

c. Divide 74317 ÷ 37. Find the Q and R.

7 4

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

2

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

7 4

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

2

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

v. The two skipped-spaces must be filled by two “0’s”.

7 4

0 0

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

2

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

v. The two skipped-spaces must be filled by two “0’s”.

7 4

80 0

One checks that the quotient is 8.

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

vi. Continue, subtract 8x37=296

2

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

v. The two skipped-spaces must be filled by two “0’s”.

7 4

80 0

2 9 6

One checks that the quotient is 8.

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

vi. Continue, subtract 8x37=296 so R=21, which is not enough to be divided by 37, so stop.

2

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

v. The two skipped-spaces must be filled by two “0’s”.

7 4

80 0

2 9 62 1

One checks that the quotient is 8.

Division

)3 7 7 4 3 1 7

i. Starting from the left, 37 goes into 74 twice.

ii. Subtract 2x37.

3 1 7iii. Bring down the rest of the digits, this is the new dividend.

vi. Continue, subtract 8x37=296 so R=21, which is not enough to be divided by 37, so stop.

2

Hence 74317 ÷ 37 = 2008 with R = 21,or that 74317 = 2008(37) + 21.

c. Divide 74317 ÷ 37. Find the Q and R.

iv. We need the entire 317 to be divided by 37.

v. The two skipped-spaces must be filled by two “0’s”.

7 4

80 0

2 9 62 1

One checks that the quotient is 8.