GCF and LCM · part 4 of 7
Finding the GCF and LCM from the Prime Factors
Line the two lists up, circle the pairs, and read both answers off.
Step 2: line them up and circle the pairs
Write one list above the other and match up the primes they both have. Each match is a pair. A prime with no partner is a leftover.
3 pairs, and 2 left over.
GCF = one from each pair = 2 x 2 x 3 = 12
LCM = one from each pair, then every leftover = 2 x 2 x 3 x 2 x 3 = 72
That picture is the whole lesson. The green pairs answer one question and the red leftovers answer the other. Everything below is just reading it.
The pairs give you the GCF
The greatest common factor is the biggest number that divides into both. A number can only divide into 24 if it is built out of 24's primes, and it can only divide into 36 if it is built out of 36's primes. So it has to be built out of the primes they share, which is exactly the pairs.
Take one number from each pair and multiply: 2 x 2 x 3 = 12.
Check it the plain way: 24 ÷ 12 = 2 and 36 ÷ 12 = 3, both exact. And nothing bigger works, because you have already used every prime the two numbers share. There is nothing left to give it.
The pairs plus the leftovers give you the LCM
The least common multiple is the smallest number that both of them divide into. That number has to contain everything 24 is made of, and everything 36 is made of. Where they overlap you only need one copy, which is the pairs. Where they differ you need both, which is the leftovers.
So take one from each pair, then every leftover: 2 x 2 x 3 x 2 x 3 = 72.
Check it: 72 ÷ 24 = 3 and 72 ÷ 36 = 2, both exact. And nothing smaller works, which is what the next picture shows.