Performing algebraic operations frequently requires factoring out multipliers that the different terms have in common. This is what is done when reducing fractions. The fraction 10/12 can be reduced, because both the numerator and denominator have factors of 2; they're both divisible by 2.

When reducing fractions or factoring out other algebraic expressions, the process is easier when you recognize the Greatest Common Factor (GCF) of the numbers. For instance, the GCF of the numbers 48 and 60 is the number 12. True, both 48 and 60 are also evenly divisible by 2 or 3 or 6, but recognizing the *greatest* possible divisor is more efficient — saving time in the long run.

To help determine the Greatest Common Factor for two or more numbers, here are some divisibility rules. These are helpful methods for recognizing when a number is divisible by another — or not.

A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.

A number is divisible by 4 if the last two digits form a number divisible by 4.

A number is divisible by 8 if the last three digits form a number divisible by 8.

A number is divisible by 5 if it ends in 0 or 5.

A number is divisible by 10 if it ends in 0.

A number is divisible by 3 if the sum of the digits is divisible by 3.

A number is divisible by 9 if the sum of the digits is divisible by 9.

A number is divisible by 6 if it's divisible by both 2 and 3.

A number is divisible by 12 if it's divisible by both 3 and 4.

A number is divisible by 15 if it's divisible by both 3 and 5.