GCF

GCF stands for Greatest Common Factor, in other words, the largest number that is a factor of a set of numbers. 

GCF is helpful for many math operations, including simplifying fractions. The fastest and most efficient way to simplify a fraction is to divide the numerator and denominator by their GCF.

How do you find a GCF?

For many numbers, if you know your times tables, it's easy to figure out the GCF, it might even be obvious.

But for bigger numbers, or if you do not know your times tables, it's helpful to use a system.

The List System:

The list system involes listing out the factors of the numbers you are trying to find the GCF for, and circling the largest factor that the lists have in common. 

Example: Find the GCF of 15 and 36 

15: 1          15

When using the list system, we like to list factors in pairs, starting from smallest (and largest) and working toward the middle.  By working number by number, in pairs, until the two sides of the list meet helps students make sure that they do not skip any factors.   The bolded line is the complete list of factors.

15: 1, 3, 5, 15

36: 1                               36

36: 1, 2,                     18, 36

36: 1, 2, 3,            12, 18, 36

36: 1, 2, 3, 4,     9, 12, 18, 36

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCF of 15 and 36 = 

15: 1, $\boxed{3}$, 5, 15

36: 1, 2, $\boxed{3}$, 4, 6, 9, 12, 18, 36

The Prime Factor System:

Another way to find the GCF is to find the prime factorization of each number, select the prime factors that the numbers have in common, and multiplying those factors together to find the greatest common factor.

Example: Find the GCF of 24 and 36

$\eqalign{2&4\\/\;&\;\backslash\\6\quad & \quad 4\\/\backslash \quad&\quad /\backslash\\\boxed{2}\quad\boxed{3}\; & \; \boxed{2} \quad \boxed{2}}$

Prime factorization of 24: 2, 2, 2, 3

$\eqalign{3&6\\/\;&\;\backslash\\9\quad & \quad 4\\/\backslash \quad&\quad /\backslash\\\boxed{3}\quad\boxed{3}\; & \; \boxed{2} \quad \boxed{2}}$

Prime factorization of 36: 2, 2, 3, 3

Prime factors in common: 2, 2, 3

$2 \times 3 \times 2 \times 2=12$

GCF of 24 and 36 is 12.

No matter what system students use, it's helpful for them to know what a GCF is (and that because it is a factor, the GCF will never be smaller than any of the numbers it's a factor of).