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Our Lesson:Our Lesson:

Review of factors

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WARM UP

(a2/a4)-3 = a 6

{(x2v2)-2.(x6s2)} / xvs = xs/v5

Is a2/ a0 = 1 / a2 ? No

Make (-3)y = -(3)y for all values of y

Multiply Left Hand Side by (-1)y+1

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A factor of a number is the exact divisor of that number which means that when the latter number is divided by the former number the remainder is zero

Factors

A factor can also be expressed as a product of two whole numbers, then these two whole numbers are called factors of that number.

The number is called a multiple (product) of each of its factors

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If we divide 20 by 5

5 is a factor of 20 as it divides exactly into 20.

20 = 1 x 20 = 2 x 10 = 4 x 5

So, the factors of 20 are 1, 2, 4, 5, 10 and 20.

Lets review it with an example

20 is the Multiple of 5

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Properties of factors

1. Every number is a factor of itself

2. Every factor of a number is an exact divisor of that number

3. 1 is a factor of every number

4. A number is the largest factor of itself or Every factor is less than or equal to the given

number

5. The number of factors of a number is finite

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Properties of Multiples

1)Every multiple of a number is greater than or equal to that number

For Example: 6 is a Multiple of each of its factors 1, 2, 3 and 6

2) The number of multiples of a number is infinite

Example: Multiples of 4 are 4, 8, 12, 16, 20, …

3) Every number is a multiple of itself Example : 1 x 14 = 14

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A number is divisible by…

… 2 if the number is even.

… 3 if the sum of the digits within the number is divisible by 3. For example, the number 12,546 is divisible by 3 because 1+2+5+4+6 or 18 is divisible by 3.

… 4 if the number formed by the last two digits of the number is divisible by 4. For example,67,984 is divisible by 4 because 84 is divisible by 4.

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Divisibility Rules

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A number is divisible by …

… 5 if the number ends with either a 0 or 5.

…6 if the number passes the divisibility tests for both 2 and 3. In other words, it must be even and the sum of its digits must be divisible by 3

… 7 if the difference of the number formed by removing the last digit and two times that last digit is divisible by 7. For example, 3794 is divisible by 7 because 379 – 2(4) or 371 is divisible by 7. If this is not clear repeat the process again using the number 371: 37 –2(1) or 35.

…

Divisibility Rules

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… 8 if the number formed by the last three digits of the number is divisible by 8. For example, 2,657,128 is divisible by 8 because 128 is divisible by 8.

… 9 if the sum of the digits within the number is divisible by 9. For example, the number 12,546 is divisible by 9 because 1+2+5+4+6 or 18 is divisible by 9.

… 10 if the number ends with 0.

Divisibility RulesA number is divisible by …

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A number is divisible by …

… 11 if the difference of the sum of every other digit within the number and the sum of the remaining digits within the number is divisible by 11. For example, the number 40,865 is divisible by 11 because (4+8+5) – (0+6) or 17 – 6 or 11 is divisible by 11.

Divisibility Rules

… 12 if the number passes the divisibility tests for both 3 and 4. In other words, the sum of the digits must be divisible by 3 and the number formed from the last two digits of the number must be divisible by 4.

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…13 If the sum of the number formed by removing the last digit and four times that last digit is divisible by 13.

For example, 7384 is divisible by 13 because 738 + 4(4) or 754 is divisible 13. If this is not clear repeat the process again using the number 754: 75 + 4(4) or 91.

If this is not clear repeat the process again using the number 91: 9 + 4(1) or 13.

A number is divisible by

Divisibility Rules

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By remembering these rules By remembering these rules we can find out factors of any we can find out factors of any numbernumberFind Factors of 108

Lets check one by one for factors…

2: yes; It’s a even number

3: yes; as 1 + 0 + 8 = 18 is divisible by 3

4:yes; as last 2 digits 08 are divisible by 4

5: No; the ones digit is not 0 or 5

6: Yes as the number is divisible by 2 and 3

7: No it is not divisible

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9: yes as the sum of digits is 18 and it is divisible by 9

10: No; As the ones digit is not 0

12: Yes as the number is divisible by 3 and 4

13: no as the sum of the number formed by removing the last digit and four times that last digit is divisible by 13.

We get the factors as 2, 3, 6, 9 and 12

Example conti…

8: No it is not divisible

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Determine whether the number is divisible by 6

1. 785 No

2. 78642 yes

3. 297144 Yes

Find the factors of

28 7, 3, 1 and 28

Lets take another example

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A prime number is a positive integer that is not the product of two smaller positive integers.

Note that the definition of a prime number doesn't allow 1

to be a prime number: 1 only has one factor, namely 1.

Prime numbers have exactly two factors,

When a number has more than two factors it is called a composite number.

Prime factorization

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Prime numbers- whole number greater than 1 Prime numbers- whole number greater than 1 that is only divisible by itself and 1.that is only divisible by itself and 1.

Composite Numbers –Composite Numbers – whole number greater whole number greater than 1 that has than 1 that has more than 2 factorsmore than 2 factors

Prime Factorization – expressing a Prime Factorization – expressing a composite number as a product of composite number as a product of prime numbersprime numbers

The numbers 0 and 1 are neither Prime or Composite

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2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67,71, 73, 79, 83, 89, 97,

Prime numbers from 1 to 100

Sieve of Eratosthenes is an easy way to find prime numbers from 1 to 100 without checking the factors of a number

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Introduced by a Greek Mathematician, Eratosthenes, the method Sieve of Eratosthenes is an easy way to find prime numbers from 1 to 100 without checking the factors of a number

Method

1)List all numbers from 1 to 100

2)Cross out 1 because it is not a prime number

3)Encircle 2 and cross out all multiples of 2, other than 2 itself

Sieve of Eratosthenes

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4) We find the next uncrossed number is 3. Encircle 3 and cross out all the multiples of 3, other than 3 itself

5) The next uncrossed number is 5 . Encircle 5 and cross out all multiples of 5, other than 5 itself

6) Continue this till all numbers are either encircled or crossed out

7) All encircled numbers are prime numbers. All crossed out numbers other than 1 are composite numbers

Steps to find prime numbers by sieve of Eratosthenes

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Using the Sieve of Eratosthenes

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Methods for finding Methods for finding Prime Factorization Prime Factorization

460460

10 * 4610 * 46

2 * 5 23 * 22 * 5 23 * 2

460 =460 = 2 * 2 * 5 * 232 * 2 * 5 * 23

Prime factors are 2² * 5 * Prime factors are 2² * 5 * 2323

Factor Tree MethodFactor Tree Method

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Note that 2, 3 and 5 are factors of 60 and all these factors are prime numbers. We call them prime factors.

When we express a number as a product of prime factors, we have actually factored it completely. We refer to this process as prime factorization

It can be written as the product 2 x 2 x 3 x 5

The number 60 is a composite number

prime factorization

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In the following expressions is prime factorization done?

45 = 5 x 9

Prime factorization has not been done, 9 has more factors

457 = 457 x 1

Prime factorization has been done

Lets see some examples

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Greatest Common Factor of two or more numbers can be defined as the greatest number that is a factor of each number

We can find the GCF by using 2 methods

Greatest Common

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List the factors of each number. Then identify the common factors. The greatest of these common factors is the GCF

Method 1

Write the prime factorization of each number. Then identify all common prime factors and find their product, to get the GCF

Method 2

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Lets take an example

Find the GCF of 56 and 42

56 = 7 x 2 x 2 x 2

42 = 7 x 3 x 2

The common prime factors are 2 and 7 so the GCF is 2 x 7 = 14

Method 2

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The exponent is sometimes referred

to as the power

46 Base Exponent

Exponents and Multiplication

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If the base (Y) is positive then the value of

43 = 4 x 4 x 4 Yn =

Yn

is positive

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If the base (Y) is negative

The value of yn depends on whether n is odd or even

(-7)1 = -7

(-7)2 = +49

(-7)3 = -343

n =1 is odd number so value of y will be negative

n =2 is even number so value of y will be positive

n =3 is odd number so value of y is negative

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Laws of multiplying Bases

Rule for multiplying bases am x an = a m + n

Product to a power (zy)n = z n x y n

Power to a Power(am)n = a m x n

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Rule for multiplying bases

am x an = a m + n

6 x 6 x 6 x 6 x 6 = 63 + 2 = 65

63

Base Exponent Expanded form Value

6 3 6 x 6 x 6 216

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Product to a power

(z y )n = z n x y n

(3x)2 = 32. x2 = 9x2

(-5m)2 = (-5)2 . m2 = 25 m2

(2y)3 = 23 . Y3 = 8 y3

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Power to a power

(xm)n = x mn

(Y3)4 = y 3.4 = y12

(n2)5 = n 2.5 = n10

(3f2)3 = 33 . f2.3 = 27 f6

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Following the rules we have learnt lets try and simplify the expressions

(-5e2f3g)2 = (-5)2e2.2 f3.2g2

= 25e4f6g2

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(11a3b2)4 . b3 = (11)4 a12b11

(m0.n7.p0) . (m6.n2) = m6n9

(25xy)0 = 1

Lets see some examples

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Laws of Dividing Bases

1. Quotient Law cm ÷ cn = c m - n

2. Power of a quotient law (z / y)n = z n / y n

Exponents and Division

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3. Negative Exponents X -1 = 1 / X

4.Power to a Power

(am / bm)n = amn / bmn

Laws of Dividing Bases

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cm ÷ cn = cm - n

If c is any non-zero number and m is a larger number than n, m > n, we can write

In symbols if c is any non-zero number, but if n > m, we get

cm ÷ cn = 1c n - m

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Power of a quotient law

In this we raise a quotient or fraction to a power

y

n zn

yn=

Here the power (n) is same so we multiply the z/y fraction ‘n’ number of times

Dividing with the same exponents

z

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-m

It indicates the reciprocal of base as a fraction (not a

negative number)

X -m = 1xm X m =1

x

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Power to a Power

(am / bm)n = amn / bmn

00 is not allowed

Anything to the power of zero is equal to

1

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[23/32]2 = 26/34 = 64/81

34/ 38 = 3-4 = 1/81

46/ 42 = 44 = 256

18-2 = 1/182 = 1/324

Solved Examples

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Your turn now

1. Find if 682 is divisible by 6 and 4 No

2. What is the fifth multiple of 8 40

3. Factorize 250 completely. 2 x 53

4. Can this be further factorized ? Yes120 = 2 x 2 x 3 x 10

5. Find the GCF of 20 and 30 5

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6. (a2bc-2)3 a-7 = b3c-6/a

7. 32a. a-1b2 = 9b2

8. 32 / 34 = 1/9

9. 4-3 / 4-5 = 16

10. t-3s4 /s-4 t3 t-6s8

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John took a loan of $300. He promised to return it in equal installments of $50. After paying two installments John was unable to pay for two months due to some accidental expenditures. He was charged an interest of $20. Find out how much does he still have to pay and in how many equal installments can he do so?

He still have to pay $225 and he can do so in 5 equal installments of $45 each

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If three sections of grade seven in Riverdale high are going to march in the annual parade and the strength of each section is 36, 24 and 28. Find out in how many equal rows can they march in?

4 Rows each

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(s2.t5)-2 /s4t-2 .

Solve the expression and find what should be multiplied with the answer to get 1 as the final answer?

Ans = s8t8

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You had a Great Lesson Today !

Be sure to practice what you have learned today