Review the meaning of GCF. The term "greatest common factor" applies to two or more numbers when considered together. While the term can apply to many numbers, it is good to focus just on sets of two. The GCF is the largest number that both numbers can be divided by (a factor) and yield a whole number result.
Think of all the factors of the numbers being examined. For example the number 8 can be divided by: 1, 2, 4, and 8. 12 can be divided by: 1, 2, 3, 4, 6, and 12. These are the factors of 8 and 12 respectively. The greatest common factor between them is 4. Many GCF problems can be solved by inspection in this manner. Sometimes the greatest common factor is one of the numbers itself; the GCF of 4 and 12 is 4.
If the GFC is not found easily by inspection, it can be found by listing only the prime factors of the numbers. Prime factors are numbers that cannot be divided by anything else and multiply together to equal the number in question. For example the prime factors of 8 are: 1, 2, 2, and 2. They multiply together, 1 x 2 x 2 x 2 = 8. The prime factors of 12 are: 1, 2, 2, and 3. Find the GCF by finding the prime factors the numbers have in common. For 8 and 12, each have 1, 2, and 2 in common. Multiplying these together yields the GCF, 1 x 2 x 2 = 4.
Use the GCF to simplify fractions by canceling out the greatest common factor between the numerator (number on top) and the denominator (number on bottom). For example, if 9/12 were to be simplified we cancel out the GCF (in this case, 3) by dividing both the numerator and denominator by it giving 3/4.
Simplify equations by canceling out the GCF of both sides, including numbers in front of variables. For example, if the equation were 3y = 9x, both sides of the equation should be divided by the GCF of 3 and 9, which is 3. If there are more parts to the equation, then find the GCF of all of them. If the equation were 8y = 12x + 4, every number should be divided by the GCF between them all, 4. This gives 2y = 3x +1.