You know how to find multiples and how to write a prime factorization. Two numbers often share factors and share multiples, and two of those shared numbers are special.
Key idea
The greatest common factor (GCF) is the biggest number that divides both numbers. The least common multiple (LCM) is the smallest number that both numbers divide into. For 12 and 18: GCF = 6, LCM = 36.
Why it works
Prime factors show why. $12 = 2 \times 2 \times 3$ and $18 = 2 \times 3 \times 3$. A common factor can only use primes that both numbers have, and the most they share is one 2 and one 3: GCF $= 2 \times 3 = 6$. A common multiple must contain each number's whole factorization, so it needs two 2s and two 3s: LCM $= 2 \times 2 \times 3 \times 3 = 36$.
Worked example: Greatest common factor
Find the GCF of 24 and 36.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- The biggest number on both lists is 12.
Answer: 12
Worked example: Least common multiple
Find the LCM of 6 and 8.
- Count by the bigger number, 8: 8, 16, 24, …
- Stop at the first one that 6 also divides: 8 no, 16 no, 24 yes.
Answer: 24
Worked example: Which one does the story need?
Hot dogs come in packs of 10 and buns in packs of 8. What is the smallest number of each you can buy to have the same number of hot dogs and buns?
- Buying packs adds up 10 at a time and 8 at a time, so we want a number that is a multiple of both.
- Multiples of 10: 10, 20, 30, 40. 40 is also a multiple of 8.
- That is 4 packs of hot dogs and 5 packs of buns.
Answer: 40 of each (the LCM).
Common mistake
Using the wrong one. Splitting things into identical groups with nothing left over needs a number that divides both amounts: a factor, the GCF. Things that repeat until they line up again need a multiple: the LCM. Quick check: the GCF can never be bigger than the smaller number, and the LCM can never be smaller than the bigger number.