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Tips on Factoring Help

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By — McGraw-Hill Professional
Updated on Sep 27, 2011

Factoring Tips - Rules of Divisibility

Factoring is a skill that is developed with practice. The only surefire way to factor numbers into their prime factors is by trial and error. There are some number facts that will make your job easier. Some of these facts should be familiar.

  • If a number is even, the number is divisible by 2.
  • If a number ends in 0 or 5, the number is divisible by 5.
  • If a number ends in 0, the number is divisible by 10.
  • If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
  • If a number ends in 5 or 0 and the sum of its digits is divisible by 3, then the number is divisible by 15.
  • If a number is even and the sum of its digits is divisible by 3, then the number is divisible by 6.
  • If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.
  • If the sum of the digits of a number is divisible by 9 and the number is even, then the number is divisible by 18.

Examples

Example 1:

126 is even and the sum of its digits is divisible by 9: 1 + 2 + 6 = 9, so

126 is divisible by 18.

Example 2:

4545 is divisible by 5 and by 9 (4 + 5 + 4 + 5 = 18 and 18 is divisible by 9).

Factoring Tips - Identifying the Prime Factors

To factor a number into its prime factors (those which have no divisors other than themselves and 1), start with a list of prime numbers (a short list can be found on the last page of this appendix). Begin with the smallest prime number and keep dividing the prime numbers into the number to be factored. It might be that a prime number divides a number more than once. Stop dividing when the square of the prime number is larger than the number. The previous list of number facts can help you ignore 2 when the number is not even; 5 when does not end in 5; and 3 when the sum of its digits is not divisible by 3.

Examples

Example 1:

120: The prime numbers to check are 2, 3, 5, 7. The list stops at 7 because 120 is smaller than 11 2 = 121.

Example 2:

249: The prime numbers to check are 3, 7, 11, 13. The list does not include 2 and 5 because 249 is not even and does not end in 5. The list stops at 13 because 249 is smaller than 17 2 = 289.

Example 3:

608: The prime numbers to check are 2, 7, 11, 13, 17, 19, 23. The list does not contain 3 because 6 + 0 + 8 = 14 is not divisible by 3 and does not contain 5 because 608 does not end in 5 or 0. The list stops at 23 because 608 is smaller than 29 2 = 841.

Example 4:

342: The prime numbers to check are 2, 3, 7, 11, 13, 17. The list does not contain 5 because 342 does not end in 5 or 0. The list stops at 17 because 342 is smaller than 19 2 = 361.

Find practice problems and solutions at Tips on Factoring Practice Problems - Set 1.

Factoring Tips - Prime Factorization

To factor a number into its prime factors, keep dividing the number by the prime numbers in the list. A prime number might divide a number more than once. For instance,

12 = 2 × 2 × 3.

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