MISSION · Number sense
Common Factors and Common Multiples
Two lists, and the number where they first agree
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Factors go into a number and multiples come out of it, so factors are a short list that stops, and multiples are a long list that never does.
Two lists that point in opposite directions
The factors of 12 are the numbers that divide it exactly:
\[1, 2, 3, 4, 6, 12\]There are six of them and then no more, because nothing bigger than 12 fits inside 12.
The multiples of 12 are what you get by multiplying it:
\[12, 24, 36, 48, 60, \dots\]That list runs forever. Every number is a factor of infinitely many numbers and a multiple of only a few.
The quickest way to list factors without missing one is to go in pairs from the outside in. For 12: \(1 \times 12\), \(2 \times 6\), \(3 \times 4\) — and once the two numbers in a pair meet or cross, you have them all.
Greatest common factor
Write out both factor lists and look at what they share.
\(12: \quad 1, \; 2, \; 3, \; 4, \; 6, \; 12\) \(18: \quad 1, \; 2, \; 3, \; 6, \; 9, \; 18\)
They share \(1, 2, 3, 6\). The biggest is 6, so \(\text{GCF}(12, 18) = 6\). Some textbooks call it the highest common factor, HCF; it is the same number.
The GCF is what you divide out to put a fraction in lowest terms. \(\frac{12}{18}\) has a GCF of 6 on both floors, and dividing both by 6 gives \(\frac{2}{3}\) in one step rather than several.
Least common multiple
Now write out both multiple lists and stop at the first agreement.
\(4: \quad 4, \; 8, \; \mathbf{12}, \; 16, \; 20, \; 24\) \(6: \quad 6, \; \mathbf{12}, \; 18, \; 24\)
They first meet at 12, so \(\text{LCM}(4, 6) = 12\). They meet again at 24, and 36, and forever after — the common multiples are themselves the multiples of the LCM.
The LCM is the common denominator you want when adding fractions. For \(\frac{1}{4} + \frac{1}{6}\) you could use 24, but 12 is smaller and means less tidying afterwards.
Which one does the question want?
Word problems rarely say “GCF” or “LCM”, so read what is being done to the numbers.
- Things are being split into equal groups — the biggest party bag, the largest square tile, the most identical bouquets — that is a GCF. You are cutting a number down.
- Things happen on repeat and you are waiting for them to line up — two buses, two blinking lights, two people jogging round a track — that is an LCM. You are counting upwards until the cycles coincide.
A bus every 9 minutes and a bus every 12 minutes both leave at 9:00. The 9-minute bus goes at 9, 18, 27, 36; the 12-minute bus at 12, 24, 36. They next leave together after 36 minutes.
The shortcut when the lists get long
Listing every factor of 84 is tedious. Prime factorization does it faster: break both numbers into primes, take the shared primes at their lowest power for the GCF and every prime at its highest power for the LCM.
There is also a check worth knowing:
\[\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b\]So once you have one, you can divide to get the other. For 12 and 18: \(6 \times \text{LCM} = 216\), giving \(\text{LCM} = 36\).
Common mix-ups
- Swapping the two words. The GCF is never bigger than the smaller number; the LCM is never smaller than the bigger one. If your answer breaks that, you have the wrong one.
- Forgetting 1 and the number itself are factors. Both always count.
- Thinking the LCM is always \(a \times b\). That works only when the two share no factors. For 4 and 6 it gives 24, which is a common multiple but not the least.
- Calling 1 a prime. It is a factor of everything, but it is not prime, so it never appears in a prime factorization.
Check yourself
Find the GCF of 12 and 18, the LCM of 4 and 6, and say when two buses running every 9 and every 12 minutes next leave together.
The shared factors of 12 and 18 are 1, 2, 3, 6, so the GCF is 6. The multiples of 4 and 6 first meet at 12. For the buses, \(\text{LCM}(9, 12) = 36\) minutes.
LEVEL 3 · REFLECT
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