How to Factor Numbers

Factoring is the process of determining if a number is evenly divisible by another number. For example, because 6 is divisible by 2 and 3, we say that two and three are factors of six. Factoring is usually required by junior high and has many applications both in advanced mathematics and everyday arithmetic.

Instructions

Difficulty: Moderate

1. Consider prime factoring. This is sometimes called complete factoring and is the process of identifying every prime number by which a given number is divisible. A prime number is any integer not evenly divisible by any integer other than itself and one. The number 2 is an exception to this rule and is defined as a prime number.
2. Determine the prime factors in an integer "n." We could do this by trying to divide "n" by every prime number between 1 and "n" (non-inclusively.) This will work but requires unnecessary effort.
3. Find the square root of "n." If "n" has a factor other than itself and 1, it also must have a factor "b." That is, n/a=b means that "a" and "b" must be factors of "n." Therefore, since ab=n, we need only consider integers less or equal to the square root of "n" to find prime factors of "n." If the square root of "n" happens to be an integer, we should immediately begin searching for factors for the square root of "n" instead.
4. Check "n" for divisibility by 2. We can quickly tell this is so when the last digit of "n" is 0, 2, 4, 6 or 8. Divisibility by 5 can be determined by a 0 or 5 in the last digit. A number is divisible by 3 if the sum of its digits is a multiple of 3. For example, we know that 105 is divisible by 3 because its digits add up to 6.
5. Observe that we can now quickly determine if a large integer "n" is divisible by any of the first 10 integers other than 7 and only need to go up to the square root of "n."