Factors

Factors

A whole number that can be divided cleanly into another whole number is called a factor of that number.

Example: Factors of 10 10 can be evenly divided by 1, as 10 ÷ 1 = 10

10 can be evenly divided by 2, as 10 ÷ 2 = 5

10 cannot be evenly divided by 3: 10 ÷ 3 = 3.333

10 cannot be evenly divided by 4: 10 ÷ 4 = 2.5

10 can be evenly divided by 5: 10 ÷ 5 = 2

10 cannot be evenly divided by 6, 7, 8, or 9

10 can be evenly divided by 10: 10 ÷ 10 = 1

The factors of 10 are 1, 2, 5, and 10.

You can also look at this the other way around: if you can multiply two whole numbers to create a third number, those two

numbers are factors of the third.

Example: Factors of 10 2 x 5 = 10, so 2 and 5 are factors of 10.

1 x 10 = 10, so 1 and 10 are also factors of 10.

You will notice that 1 and the number itself are always factors of a given number.

A Note About Negatives

Everything said above also applies to negative whole numbers.

The factors of 10 are actually –1, 1, –2, 2, –5, 5, –10, and 10.

(–1 x –10 = 10, and –2 x –5 = 10.)

The factors of –10 are also –1, 1, –2, 2, –5, 5, –10, and 10.

(2 x –5 = –10, –2 x 5 = –10, and so on.)

This can be written more easily by using a combined + and – sign (±) to indicate that both the positive and negative

versions of a number are factors. Thus, the factors of 10 can be written as ±1, ±2, ±5, and ±10.

Prime Factors Some numbers can be evenly divided only by 1 and themselves. These are prime numbers. Factors that are prime numbers are called prime factors.

Every whole number greater than one is either a prime number, or can be described as a product of prime factors.

Examples: 10 is the product of the prime factors 2 x 5

11 is a prime number

12 is the product of the prime factors 2 x 2 x 3

324 is the product of the prime factors 2 x 2 x 3 x 3 x 3 x 3

700 is the product of the prime factors 2 x 2 x 5 x 5 x 7

701 is a prime number

2103 is the product of the prime factors 3 x 701

To find the prime factors of a given number, follow these steps:

1. See if the number is a prime number. If it's below 1000, use the table of prime numbers. If it is prime, add it to the

list of prime factors, and you're done.

2. If it's not prime, try dividing it by a prime number, starting with 2.

If it divides cleanly, with no remainder, then add that prime to the list of prime factors. Take the quotient as your

new number to work with, and return to step 1.

If it does not divide cleanly, return to step 2, but move on to the next prime on the list.

Example: Prime factors of 700 700 ÷ 2 = 350, with no remainder. Add 2 to the list of prime factors.

350 ÷ 2 = 175, with no remainder. Add 2 to the list of prime factors.

175 ÷ 2 = 87.5. It doesn't divide cleanly, so we go to the next prime number.

175 ÷ 3 = 58.33. It doesn't divide cleanly, so we go to the next prime number.



7 is a prime number. Add 7 to the list of prime factors, and we're done.

The prime factors of 700 are 2 x 2 x 5 x 5 x 7.

Be sure to check at each step to see if the number you have is a prime. The next example illustrates why:

Example: Prime factors of

2103

2103 ÷ 2 is 1051.5. It doesn't divide cleanly, so we go to the next prime number.

2103 ÷ 3 is 701, with no remainder. Add 3 to the list of prime factors.

The table of prime numbers will tell you that 701 is a prime. Add 701 to the list of prime factors,

and we're done.

The prime factors of 2103 are 3 x 701.

If we didn't notice that 701 was a prime, we'd have gone on to check 5, 7, 11, 13, and so on, going through 120 more primes

before getting done. So be sure to check the quotient every time before proceeding.

(As you might imagine, this method is designed for smaller numbers. It's too time-consuming for very large ones.)

Common Factors

Factors that two numbers have in common are called the common factors of those numbers.

Example:

What are the common factors of 20 and 25?

The factors of 20 are 1, 2, 4, 5, and 20.

The factors of 25 are 1, 5, and 25.

The common factors of 20 and 25 are 1 and 5.

Example:



The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

The common factors of 15 and 30 are 1, 3, 5, and 15.

Example:



The factors of 20 are 1, 2, 4, 5, and 20.

The only common factor of 9 and 20 is 1.

Greatest Common Factors

The largest common factor of two numbers is called their greatest common factor or highest common factor.

Example:

What is the greatest common factor of 20 and 25?

The common factors of 20 and 25 are 1 and 5. (see above)

The greatest common factor is 5.

Example:


The common factors of 15 and 30 are 1, 3, 5, and 15.


Example:


The only common factor of 9 and 20 is 1.


Method #2: Prime Factors

Another method of finding the greatest common factor uses the prime factors of each number. Rather than looking at all the

factors—prime or not—to find the single highest one they have in common, we multiply all the prime factors they have in

common, to reach the same results.

Example:


The prime factors of 20 are 2 x 2 x 5.


The prime factor they have in common is 5.


Example:




The prime factors they have in common are 3 x 5.

3 x 5 = 15.


When two numbers have no prime factors in common, their greatest common factor is 1.

Example:




9 and 20 have no prime factors in common.


Lowest Common Multiples

Common multiples are multiples that two numbers have in common. These can be useful when working with fractions and

ratios.

Example:

What are some common multiples of 2 and 3?

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...

Common multiples of 2 and 3 include 6, 12, 18, and 24.

Example:

What are some common multiples of 25 and 30?

Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325...

Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330...

Common multiples of 25 and 30 include 150 and 300.

The lowest common multiple or least common multiple is the lowest multiple two numbers have in common.

There are two ways of finding the lowest common multiple of two numbers.

Method 1: Listing Multiples

The first way to find the lowest common multiple is to do what we did above: write out a list of the lowest multiples of each

number, and look for the lowest multiple both numbers have in common.

Example:

What is the lowest common multiple of 2 and 3?

Multiples of 2: 2, 4, 6, 8...

Multiples of 3: 3, 6, 9...

The lowest common multiple of 2 and 3 is 6.

Example:


Multiples of 25: 25, 50, 75, 100, 125, 150, 175...

Multiples of 30: 30, 60, 90, 120, 150, 180...


Method 2: Factors

The other way to find the lowest common multiple is to list the prime factors for each number. Remove the prime factors

both numbers have in common. Multiply one of the numbers by the remaining prime factors of the other number. The result

will be the lowest common multiple.

Example:




Remove the 5 that 25 and 30 have in common as a prime factor.

Multiply 25 by the remaining prime factors of 30.

25 x 2 x 3 = 150.


You'll get the same results no matter which number you work with:

Example:






30 x 5 = 150.


Another Example

Example:



The prime factors of 48 are 2 x 2 x 2 x 2 x 3.

Remove the 2 x 3 that 42 and 48 have in common as prime factors.


48 x 7 = 336.


What if they have no prime factors in common?

Example:





Either:


44 x 3 x 3 x 5 = 1980.

Or:


45 x 2 x 2 x 11 = 1980.

Or:

44 x 45 = 1980.


As that last example illustrates, if two numbers have no prime factors in common, the lowest common multiple will be equal

to the product of the two numbers.

Treat primes as prime factors

If one number is prime, you can treat it as its own prime factor.

Example:


7 is a prime number.



7 x 30 = 210.


Example:





2 x 3 = 6.


Example:





Either:


3 x 2 x 5 = 30

Or:

You would normally multiply 30 by the remaining prime factors of 3, but there are no remaining prime factors.

The lowest common multiple is 30.

Prime results

As you can see from the above, there are two scenarios if at least one number is prime:

If one number is prime, and the other number's prime factors include that prime number, the lowest common

multiple will be equal to the non-prime number.

If one number is prime, and the other number's prime factors do not include that prime number, the lowest

common multiple will be equal to the product of the two numbers.

(The second scenario also includes cases where both numbers are prime.)

Mixed Numbers and Improper Fractions

A mixed number is a combination of a whole number and a fraction. For example, if you have two whole apples and one

half apple, you could describe this as 2 + 1/2 apples, or 21/2 apples.

Writing Mixed Numbers as Fractions

This mixed number can also be expressed as a fraction. Each whole apple contains two half apples. Your two whole apples

are also four half apples. Four half apples plus one half apple is five half apples. So you have 5/2 apples.

To put this another way: to turn a mixed number into a fraction, multiply the whole number by the denominator (the

bottom part), and add the result to the numerator (the top part).

21/2 = ?

Multiply the whole number by the denominator.

The whole number is 2.

The denominator is 2.

2 x 2 = 4.

Add the result to the numerator:

The numerator is 1.

4 + 1 = 5

The numerator is 5. The denominator remains 2.

21/2 = 5/2

Another Example

Let's try another example:

52/3 = ?

Multiply the whole number by the denominator.

The whole number is 5.

The denominator is 3.

5 x 3 = 15.

Add the result to the numerator:

The numerator is 2.

15 + 2 = 17

The numerator is 17. The denominator remains 3.

52/3 = 17/3

Proper and Improper Fractions

A fraction in which the numerator is smaller than the denominator, like 1/3 or 2/5 is called a proper fraction. A fraction in

which the numerator is larger than or equal to the denominator, like 5/2, 17/3, or 6/6 is called an improper fraction. (To put it

another way, a fraction with a value less than 1 is a proper fraction. A fraction with a value greater than or equal to 1 is an

improper fraction.)

As we have shown above, mixed numbers can be written as improper fractions. Similarly, improper fractions can be written

as mixed numbers.

Writing Improper Fractions as Mixed Numbers

To write an improper fraction as a mixed number, divide the numerator (top part) by the denominator (bottom part).

The quotient is the whole number, and the remainder is the numerator.

How would you express 17/4 as a mixed number?

Divide the numerator by the denominator:

17 ÷ 4 = 4, with a remainder of 1

The quotient, 4, is the whole number. The remainder, 1, is the numerator. The denominator remains 4.17/4 = 41/4

Two More Examples

Let's try another couple of examples:

14/9 = ?


14 ÷ 9 = 1, with a remainder of 5

The quotient, 1, is the whole number. The remainder, 5, is the numerator. The denominator remains 9.14/9 = 15/9

If there is no remainder, just take the quotient as the whole number:

20/5 = ?


20 ÷ 5 = 4

The quotient, 4, is the whole number. There is no remainder.20/5 = 4

For more fun and practice with mixed numbers and improper fractions, see the Fraction Cafe!

Multiplying Fractions and Mixed Numbers

Multiplying Fractions

If your friend has one-quarter of a pie, and she gives you half, how much of the pie do you have? Or, to put it another way,

what's half of one-quarter? Or, to put it into mathematical notation:

1/2 x 1/4 = ?

To get the answer, multiply the numerators (the top parts) and denominators (the bottom parts) separately.

In this case, first we multiply the numerators:

1 x 1 = 1

Next we multiply the denominators:

2 x 4 = 8

The answer has a numerator of 1 and a denominator of 8. In other words:

1/2 x 1/4 = 1 x 1/2 x 4 = 1/8

You have one-eighth of the pie.

Another Example

Let's try another.

2/9 x 3/4 = ?

First we multiply the numerators:

2 x 3 = 6


9 x 4 = 36


2/9 x 3/4 = 2 x 3/9 x 4 = 6/36

This can be further reduced:

6 ÷ 6/36 ÷ 6 = 1/6

(See Reducing Fractions.)

Multiplying Mixed Numbers

To multiply two mixed numbers, or a mixed number and a fraction, first convert each mixed number to a fraction. Then

multiply the fractions.

What is 21/3 x 1/4 = ?

First we write 21/3 as a fraction:

21/3 = 7/3

Then we multiply the fractions.

7/3 x 1/4 = ?

First we multiply the numerators:

7 x 1 = 7


3 x 4 = 12


21/3 x 1/4 = 7 x 1/3 x 4 = 7/12

Reciprocal Fractions

To find the reciprocal of a fraction, flip it over, so that the numerator becomes the denominator and the denominator

becomes the numerator. That is:

The reciprocal of 4/5 is 5/4

Note that the product of a fraction and its reciprocal is always 1.

4/5 x 5/4 = 20/20 = 1

In the case of a whole number, think of it as having a denominator of 1:

The reciprocal of 5 is 1/5.5/1 x 1/5 = 5/5 = 1

Every number has a reciprocal except for 0. There is nothing you can multiply by 0 to create a product of 1, so it has no

reciprocal.

Reciprocals are used when dividing fractions.

Dividing Fractions

If your friend has half a pie, how many quarter-pies are in that half? Or, to put this into mathematical notation:

1/2 ÷ 1/4 = ?

To get the answer, flip the divisor (the second fraction) over, and then multiply the fractions. (Or, to put it another way,

multiply the dividend [the first fraction] by the reciprocal of the divisor [the second fraction].)

In this case, that makes the problem:

1/2 x 4/1 = ?

We begin by multiplying the numerators:

1 x 4 = 4

And then we multiply the denominators:

2 x 1 = 2


1 x 4/2 x 1 =4/2

This fraction can be reduced to lowest terms:

4 ÷ 2/2 ÷ 2 =2/1 = 2

There are 2 quarter-pies in a half-pie.

Another Example

Let's try another:

4/5 ÷ 6/7 = ?

We flip the divisor over, and change the division sign to a multiplication sign:

4/5 x 7/6 = ?

We multiply the numerators:

4 x 7 = 28

And we multiply the denominators:

5 x 6 = 30


4 x 7/5 x 6 =28/30

We can reduce this fraction by dividing the numerator and denominator by 2:

28 ÷ 2/30 ÷ 2 = 14/15

Mixed Numbers

Let's try one more, this time with a mixed number:

21/4 ÷ 2/3 = ?

First we change the mixed number to an improper fraction:

9/4 ÷ 2/3 = ?

Next we flip the divisor over and change the division sign to a multiplication sign:

9/4 x 3/2 = ?

We multiply the numerators:

9 x 3 = 27

And we multiply the denominators:

4 x 2 = 8


9 x 3/4 x 2 =27/8

Finally, we turn the result—an improper fraction—into a mixed number.

27/8 = 33/8 =

Reducing Fractions to Lowest Terms

Consider the following two fractions:

1/2 and 2/4

These fractions are equivalent fractions. They both represent the same amount. One half of an orange is equal to two

quarters of an orange. However, only one of these fractions is written in lowest terms.

A fraction is in lowest terms when the numerator and denominator have no common factor other than 1.

The factors of 2 are 1 and 2.


2 and 4 share a common factor: 2.

We can reduce this fraction by dividing both the numerator and denominator by their common factor, 2.

2 ÷ 2/4 ÷ 2 = 1/2

1 and 2 have no common factor other than 1, so the fraction is in lowest terms.

Method #1: Common Factors

(a slow and steady method)

Let's try another example:

30/36

Do 30 and 36 share any factors other than 1?

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

30 and 36 have three common factors: 2, 3, and 6.

Let's see what happens if we divide the numerator and denominator by their lowest common factor, 2. (In fact, we'd know

that they have 2 as a common factor without having to work out all their factors, because both 30 and 36 are even numbers.)

30 ÷ 2/36 ÷ 2 = 15/18

Are we done? Do 15 and 18 share any factors other than 1?

The factors of 15 are 1, 3, 5, 15.

The factors of 18 are 1, 2, 3, 6, 9, 18.

15 and 18 have one common factor: 3.

Once again, we divide the numerator and denominator by their common factor, 3.

15 ÷ 3/18 ÷ 3 = 5/6

Are we done? Do 5 and 6 share any factors other than 1?

The factors of 5 are 1 and 5.


5 and 6 have no common factors other than 1.

This method will reduce a fraction to its lowest terms, but it can take several steps until you reach that point. What would

have happened if, instead of dividing the numerator and denominator by their lowest common factor, we had started with

their greatest common factor?

Method #2: Greatest Common Factor

(a more efficient method)

Let's try it again:

30/36

Do 30 and 36 share any factors other than 1?

The factors of 30 are 1, 2, 3, 5, 6, 10, 15.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18.

30 and 36 have three common factors: 2, 3, and 6.


Divide the numerator and denominator by the greatest common factor:

30 ÷ 6/36 ÷ 6 = 5/6

This time, it takes only one step to get to the same result. To reduce a fraction to its lowest terms, divide the numerator

and denominator by the greatest common factor.

Method #3: Prime Factors

(an even more efficient method)

Another way to reduce fractions is to break the numerator and denominator down to their prime factors, and remove every

prime factor the two have in common. Let's do that example one more time, using this method.

30/36


The prime factors of 36 are 2 x 2 x 3 x 3.2 x 3 x 5/2 x 2 x 3 x 3

We remove the 2 x 3 the numerator and denominator have in common:5/2 x 3 = 5/6

(If you think about it, this works the same way as the last method. The greatest common factor of two numbers is the same

as the product of the prime factors they have in common.)

For more fun and practice with fractions, see the Fraction Cafe!

Roman Numerals

Try the Roman Numeral Challenge.

Roman numerals are expressed by letters of the alphabet:

I=1

V=5

X=10

L=50

C=100

D=500

M=1000

There are four basic principles for reading and writing Roman numerals:

1. A letter repeats its value that many times (XXX = 30, CC = 200, etc.). A letter can only be repeated three times.

2. If one or more letters are placed after another letter of greater value, add that amount.

VI = 6 (5 + 1 = 6)

LXX = 70 (50 + 10 + 10 = 70)

MCC = 1200 (1000 + 100 + 100 = 1200)

3. If a letter is placed before another letter of greater value, subtract that amount.

IV = 4 (5 – 1 = 4)

XC = 90 (100 – 10 = 90)

CM = 900 (1000 – 100 = 900)

Several rules apply for subtracting amounts from Roman numerals:

o a. Only subtract powers of ten (I, X, or C, but not V or L)

For 95, do NOT write VC (100 – 5).

DO write XCV (XC + V or 90 + 5)

o b. Only subtract one number from another.

For 13, do NOT write IIXV (15 – 1 - 1).

DO write XIII (X + I + I + I or 10 + 3)

o c. Do not subtract a number from one that is more than 10 times greater (that is, you can subtract 1 from

10 [IX] but not 1 from 20—there is no such number as IXX.)

For 99, do NOT write IC (C – I or 100 - 1).

DO write XCIX (XC + IX or 90 + 9)

4. A bar placed on top of a letter or string of letters increases the numeral's value by 1,000 times.

XV = 15, = 15,000

One I Eleven XI Thirty XXX

Two II Twelve XII Forty XL

Three III Thirteen XIII Fifty L

Four IV Fourteen XIV Sixty LX

Five V Fifteen XV Seventy LXX

Six VI Sixteen XVI Eighty LXXX

Seven VII Seventeen XVII Ninety XC

Eight VIII Eighteen XVIII One hundred C

Nine IX Nineteen XIX Five hundred D

Ten X Twenty XX One thousand M

Factors

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