Prime FactorizationPrime Factorization

77thth Grade Math Grade Math

Prime Factorization Of a NumberPrime Factorization Of a Number

A A prime numberprime number is a counting number that only is a counting number that only has two factors, itself and one. Counting numbers has two factors, itself and one. Counting numbers which have more than two factors (such as six, which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be whose factors are 1, 2, 3 and 6), are said to be composite numberscomposite numbers. . When a composite When a composite number is written as a product of all of its prime number is written as a product of all of its prime factors, we have the prime factorization of the factors, we have the prime factorization of the number. number.

There are several different methods that can be There are several different methods that can be utilized for the prime factorization of a number. utilized for the prime factorization of a number.

Using Division Using Division

Prime factors can Prime factors can be found using be found using division. division.

Keep dividing until Keep dividing until

you have all prime you have all prime numbers. The numbers. The prime factors of 78 prime factors of 78 are are 2, 3, 13.2, 3, 13.

2

133

3978

39

Remember the Divisibility Rules Remember the Divisibility Rules

If the last digit is even, the number is If the last digit is even, the number is divisible by 2. divisible by 2.

If the last digit is a 5 or a 0, the number is If the last digit is a 5 or a 0, the number is divisible by 5.divisible by 5.

If the number ends in 0, it is divisible by If the number ends in 0, it is divisible by 10.10.

If the sum of the digits is divisible by 3, the If the sum of the digits is divisible by 3, the number is also. number is also.

If the last two digits form a number If the last two digits form a number divisible by 4, the number is also. divisible by 4, the number is also.

More divisibility rules…More divisibility rules… If the number is divisible by both 3 and 2, If the number is divisible by both 3 and 2,

it is also divisible by 6. it is also divisible by 6. Take the last digit, double it, and subtract Take the last digit, double it, and subtract

it from the rest of the number; if the it from the rest of the number; if the answer is divisible by 7 (including 0), then answer is divisible by 7 (including 0), then the number is also. the number is also.

If the last three digits form a number If the last three digits form a number divisible by 8, then the whole number is divisible by 8, then the whole number is also divisible by 8. also divisible by 8.

If the sum of the digits is divisible by 9, the If the sum of the digits is divisible by 9, the number is also. number is also.

Using the Factor Tree Using the Factor Tree

7878

/ \/ \

/ \/ \

22 x x 3939

/ / \/ / \

/ / \ / / \

22 x x 3 3 xx 13 13

ExponentsExponents

7272

/ \/ \

8 x 98 x 9

/ \ / \/ \ / \

22 x 4 x x 4 x 3 3 x x 33

// / \ \ \/ \ \ \

22 x x 22 x x 22 x x 33 x x 33

Another key idea in writing Another key idea in writing the prime factorization of a the prime factorization of a number is an number is an understanding of understanding of exponentsexponents.. An exponent An exponent tells how many times the tells how many times the base is used as a factor. base is used as a factor.

72 = 272 = 23 3 x 3 x 322

Let’s Try a Factor Tree!Let’s Try a Factor Tree!

8484 / \/ \

2 x 422 x 42 / / \/ / \ 2 x 2 x 212 x 2 x 21 / / / \/ / / \ 2 x 2 3 x 7 2 x 2 3 x 7 What is the final factorization?What is the final factorization?

222 2 x 3 x 7 = 84x 3 x 7 = 84

Factor Trees do not look the same for the same number, Factor Trees do not look the same for the same number,

but the final answer is the same.but the final answer is the same. 7272

/ \/ \

8 x 98 x 9

/ \ / \/ \ / \

22 x 4 x x 4 x 3 3 x x 33

/ \/ \

22 x x 22 x x 22 x x 33 x x 33

7272

/ \/ \

22 x 36 x 36

/ / \ / / \

22 x x 22 x 18 x 18

/ / / \/ / / \

2 2 x x 2 2 x x 2 2 x 9 x 9

/ / / / \/ / / / \

22 x x 2 2 x x 22 x x 33 x x 3 3

Greatest Common Factors Greatest Common Factors

One method to find greatest common One method to find greatest common factors is to list the factors of each factors is to list the factors of each number. The largest number is the number. The largest number is the greatest common factor. greatest common factor.

Let’s find the factors of 72 and 84. Let’s find the factors of 72 and 84. 7272 1, 2, 3, 4, 6, 8, 9, 1, 2, 3, 4, 6, 8, 9, 12,12, 18, 24, 36, 72 18, 24, 36, 72

8484

1, 2, 3, 4, 6, 1, 2, 3, 4, 6, 12,12, 14, 21, 28, 42, 84 14, 21, 28, 42, 84

Prime Factorization is helpful for finding Prime Factorization is helpful for finding

greatest common factors.greatest common factors. 7272

/ \/ \

8 x 98 x 9

/ \ / \/ \ / \

22 x 4 x x 4 x 3 3 x x 33

/ \/ \

22 x x 22 x x 22 x x 33 x x 33

Take the common prime Take the common prime factors of each number factors of each number and multiply to find the and multiply to find the greatest common factor. greatest common factor.

8484

/ \/ \

22 x 42 x 42

/ / \/ / \

2 x 22 x 2 x 21 x 21

/ / / \/ / / \

2 2 x x 2 2 33 x 7 x 7

2 x 2 x 3 = 122 x 2 x 3 = 12

ResourcesResources

IXL.Com – Sixth Grade – N.5 Prime IXL.Com – Sixth Grade – N.5 Prime Factorization Factorization

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