This week in math 10 we learned how to find the GFC and LCM of two or more numbers.

GFC (Greatest Common Multiple):

First, find the prime factorization of your numbers. Then take all of your common prime factors and multiply them together to find your GCF. If common factors have exponents, take the lesser of the two exponents when you go to multiply the common factors together.

Example: Take your two numbers (48 and 72 here) and find the prime factorization.

(prime factorization)

Next, you take the prime factors (circled above) and find common factors. Take whichever common factors have the LOWEST exponent, and multiply the result together like so:

Therefore the GCF of 48 and 72 is 24.

LCM (Lowest Common Multiple):

To begin, once again find the prime factorization of your numbers. Find all of the prime factors as you did with the GCF, however instead of only taking common factors, you take ALL the prime factors. If two or more prime factors are the same, you take the one with the GREATEST exponent this time. Then, simply multiply all the factors together to get the LCM.

Example: Take your numbers (18 and 63 here) and find the prime factorization.

(prime factorization)

Next, take all prime factors and multiply them. Keeping in mind if two factors are the same, you are taking the one with the lowest exponent.

(Both 18 and 63 had a prime factor of 3, but instead of taking both, only one was taken. Exponents were the same so were not a factor in the decision.)

Therefore the LCM of 18 and 63 is 126.