Prime Factorization Applications

Prime Factorization Applications

by Mrs. Lown

​Number of Factors

​To find the number of factors a number has:

• Find the prime factorization

• ​Write it in exponent form

• ​Add one to each exponent and multiply exponents together

​

​Example:

​ How many factors does 24 have?

​ 24 = 2³3¹

​ The exponents are 3 and 1. (3 + 1) * (1 + 1) = 4(2) = 8 factors

​ 1, 2, 3, 4, 6, 8, 12, 24 <- There are 8 factors!

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How many factors does 65 have?

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How many factors does 84 have?

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How many factors does 480 have?

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Find the number of factors for  $13^931^4$  

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How many factors does 630 have?

​Sum of Factors

​To find the sum of all the factors of a number:

• Find the prime factorization

• ​Find the sum of the powers

• ​Multiply the sums

​

​Example:

​Find the sum of factors for 12

​12 = 2²3¹ = (2⁰ + 2¹ + 2²) * (3⁰ + 3¹) = (1 + 2 + 4) x (1 + 3) = 7 x 4 = 28

​

​Check: factors of 12 are 1, 2, 3, 4, 6, 12. 1 + 2 + 3 + 4 + 6 + 12 = 28!!

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Find the sum of all the factors for 30.

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Find the sum of all the factors for 36.

​Practice

​Now here are some higher-level questions dealing with prime factorization!

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Find the sum of the factors that are not factors of 30 but are factors of 60.

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How many more factors does 60 have than 30?

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Find the positive difference between the number of factors and the sum of all the factors for 210.

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Challenge: Find the number of factors for 12,000