Prime Factorization 7 th Grade Math

Prime Factorization Of a Number  A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be composite numbers. When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number.  There are several different methods that can be utilized for the prime factorization of a number.

Using Division  Prime factors can be found using division.  Keep dividing until you have all prime numbers. The prime factors of 78 are 2, 3, 13.

Remember the Divisibility Rules IIIIf the last digit is even, the number is divisible by 2. IIIIf the last digit is a 5 or a 0, the number is divisible by 5. IIIIf the number ends in 0, it is divisible by 10. IIIIf the sum of the digits is divisible by 3, the number is also. IIIIf the last two digits form a number divisible by 4, the number is also.

More divisibility rules… IIIIf the number is divisible by both 3 and 2, it is also divisible by 6. TTTTake the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also. IIIIf the last three digits form a number divisible by 8, then the whole number is also divisible by 8. IIIIf the sum of the digits is divisible by 9, the number is also.

 Using the Factor Tree / \ / \ 2 x 39 2 x 39 / / \ / / \ 2 x 3 x 13

Exponents / \ / \ 8 x 9 8 x 9 / \ / \ / \ / \ 2 x 4 x 3 x 3 2 x 4 x 3 x 3 / / \ \ \ / / \ \ \ 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 3 x 3  Another key idea in writing the prime factorization of a number is an understanding of exponents. An exponent tells how many times the base is used as a factor. 72 = 2 3 x 3 2

Let’s Try a Factor Tree! / \ / \ 2 x 42 2 x 42 / / \ / / \ 2 x 2 x 21 2 x 2 x 21 / / / \ / / / \ 2 x 2 3 x 7 2 x 2 3 x 7 What is the final factorization? 2 2 x 3 x 7 = 84

Factor Trees do not look the same for the same number, but the final answer is the same / \ / \ 8 x 9 8 x 9 / \ / \ / \ / \ 2 x 4 x 3 x 3 2 x 4 x 3 x 3 / \ / \ 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 3 x 372 / \ / \ 2 x 36 2 x 36 / / \ / / \ 2 x 2 x 18 2 x 2 x 18 / / / \ / / / \ 2 x 2 x 2 x 9 2 x 2 x 2 x 9 / / / / \ / / / / \ 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 3 x 3

Greatest Common Factors  One method to find greatest common factors is to list the factors of each number. The largest number is the greatest common factor.  Let’s find the factors of 72 and , 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, , 2, 3, 4, 6, 12, 14, 21, 28, 42, 84

Prime Factorization is helpful for finding greatest common factors / \ / \ 8 x 9 8 x 9 / \ / \ / \ / \ 2 x 4 x 3 x 3 2 x 4 x 3 x 3 / \ / \ 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 3 x 3 Take the common prime factors of each number and multiply to find the greatest common factor / \ / \ 2 x 42 2 x 42 / / \ / / \ 2 x 2 x 21 2 x 2 x 21 / / / \ / / / \ 2 x 2 3 x 7 2 x 2 3 x 7 2 x 2 x 3 = 12

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