Lowest Common Multiple

List the multiples of 4: 4, 8, 12, 16, 20, 24… and the multiples of 6: 6, 12, 18, 24… The first number that appears in both lists is 12. That makes it the lowest common multiple.

A common mistake is to simply multiply: \(4 \times 6 = 24\). But 24 is a common multiple, not the lowest one. Because 4 and 6 share a common factor of 2, their LCM is less than their product.

Multiples of 3: 3, 6, 9, 12… Multiples of 9: 9, 18, 27… The number 9 appears in both lists, and it is the smallest such number. So the LCM is 9 — one of the original numbers.

This always happens when one number is a multiple of the other: if \(b\) is a multiple of \(a\), then the LCM is simply \(b\). Students sometimes feel the LCM should be “bigger than both” or a completely new number, but that’s not necessarily the case.

Multiples of 8: 8, 16, 24, 32… Multiples of 12: 12, 24, 36… The first number in both lists is 24. The product \(8 \times 12 = 96\) is much larger — and certainly a common multiple — but 24 beats it as the lowest.

Using prime factorisations: \(8 = 2^3\) and \(12 = 2^2 \times 3\). The LCM takes the highest power of each prime: \(2^3 \times 3 = 24\). This method is especially useful when listing multiples would take a long time.

The common multiples of 3 and 5 are: 15, 30, 45, 60, 75… Every one of these is a multiple of 15. This is no coincidence: 15 is the LCM of 3 and 5, and all common multiples of two numbers are always multiples of their LCM.

Think of it this way: if a number is divisible by both 3 and 5, it must contain at least one factor of 3 and one factor of 5 — which means it must be divisible by \(3 \times 5 = 15\). The LCM acts as a “building block” for all common multiples.

Example: 4 and 6

Another: 3 and 12

Creative: 4 and 12 — students often forget that one number can be a factor of the LCM. Or 1 and 12 — since LCM(1, n) is always n.

Trap: 2 and 6 — a student might think \(2 \times 6 = 12\) so the LCM must be 12. But the actual LCM(2, 6) = 6, not 12. This is the classic “LCM equals the product” misconception.

Example: 5 and 15 — since 15 is a multiple of 5, LCM(5, 15) = 15.

Another: 7 and 21 — since 21 is a multiple of 7, LCM(7, 21) = 21.

Creative: 1 and 17 — every number is a multiple of 1, so LCM(1, n) is always n. Or 6 and 6 — LCM of a number with itself is that number.

Trap: 6 and 9 — a student might think “9 is the bigger number, so LCM is 9.” But 9 is not a multiple of 6. The actual LCM(6, 9) = 18, not 9. The LCM only equals the larger number when the larger is a multiple of the smaller.

Example: 4 and 9 — LCM = 36 = \(4 \times 9\).

Another: 7 and 8 — LCM = 56 = \(7 \times 8\).

Creative: 8 and 15 — both composite, but they share no common factor (coprime), so LCM = \(8 \times 15 = 120\). Pairs of primes like 2 and 11 work too, but two composite coprime numbers is a less obvious choice.

Trap: 6 and 10 — a student might assume any two numbers work. But 6 and 10 share a common factor of 2, so LCM(6, 10) = 30, not 60. The LCM equals the product only when the numbers are coprime (HCF = 1).

Example: 40 — it’s divisible by both 4 and 10, but the LCM(4, 10) = 20, so 40 is a common multiple that isn’t the lowest.

Another: 60

Creative: 2000 — an extremely large common multiple, far from the LCM. Or 100 — since \(100 \div 4 = 25\) and \(100 \div 10 = 10\), it works, and it’s a number students wouldn’t immediately associate with 4 and 10.

Trap: 20 — a student might offer this, but 20 is the LCM of 4 and 10. The question asks for a common multiple that is not the lowest, so 20 doesn’t qualify. This tests whether students understand the difference between “a common multiple” and “the lowest common multiple.”

Example: 2, 3, and 8

Another: 4, 6, and 8

Creative: 3, 8, and 12

Trap: 2, 4, and 12 — a student might see that 24 is a multiple of all three and stop there. But the actual LCM(2, 4, 12) = 12, not 24. Just because a number works as a common multiple doesn’t make it the lowest.

True when the numbers are coprime (share no common factor other than 1): LCM(3, 5) = 15 = \(3 \times 5\). Also true for any pair of distinct primes, like LCM(2, 7) = 14.

False when the numbers share a common factor: LCM(9, 12) = 36, but \(9 \times 12 = 108\). The shared factor of 3 means the LCM is much less than the product. In general, \(\text{LCM}(a,b) = \frac{a \times b}{\text{HCF}(a,b)}\), so the LCM equals the product only when HCF = 1.

True when neither number is a multiple of the other: LCM(3, 7) = 21, which is greater than both 3 and 7.

False when the larger number is a multiple of the smaller: LCM(5, 15) = 15, which equals the larger number, not greater. The LCM is always at least as large as both numbers, but not necessarily strictly greater.

Consecutive whole numbers (like 7 and 8, or 11 and 12) are always coprime — their only common factor is 1. Any common factor of \(n\) and \(n+1\) must divide their difference, which is 1. So HCF = 1, which means LCM = product.

Try a few: LCM(3, 4) = 12 = \(3 \times 4\). LCM(9, 10) = 90 = \(9 \times 10\). LCM(99, 100) = 9900 = \(99 \times 100\). It works every time.

The LCM is a multiple of both numbers. A multiple of a positive whole number is always greater than or equal to that number. So the LCM can never be smaller than either of the two numbers.

This catches students who confuse LCM with HCF. The highest common factor is always less than or equal to both numbers, but the lowest common multiple is always greater than or equal to both numbers. They work in opposite directions.

If you multiply the inputs by a scale factor, the common multiples scale by the exact same amount. This builds deep structural number sense!

For example, LCM(3, 4) = 12. If we double both numbers to 6 and 8, the new LCM is 24 (which is double 12). This works for any scale factor.

The student has assumed that “the LCM is always the larger number.” This overgeneralises from cases like LCM(3, 9) = 9 or LCM(5, 20) = 20, where one number happens to be a multiple of the other. The rule only works in that special case — when the larger number is a multiple of the smaller.

Here, 10 is not a multiple of 8, so the LCM must be larger than both. Multiples of 8: 8, 16, 24, 32, 40… Multiples of 10: 10, 20, 30, 40… The correct answer is 40.

The answer is correct — LCM(5, 7) is 35 — but the reasoning is dangerously wrong. The student claims that “multiply the numbers” always gives the LCM. It worked here only because 5 and 7 are coprime (they share no common factor other than 1).

A single counterexample breaks the rule: LCM(6, 8) is 24, not \(6 \times 8 = 48\). If the student relies on “just multiply” they will get the wrong answer whenever the numbers share a common factor. Getting the right answer with faulty reasoning is more dangerous than getting it wrong — the student doesn’t know their method has a limit.

The student has found the highest common factor (HCF), not the lowest common multiple (LCM). This is the classic “confusing factors and multiples” error. They listed factors and found the biggest shared one — that’s the HCF procedure, not the LCM procedure.

For the LCM, they should list multiples: multiples of 12 are 12, 24, 36, 48… and multiples of 18 are 18, 36, 54… The first common multiple is 36. Notice 36 is much larger than 6 — the LCM is always ≥ both numbers, while the HCF is always ≤ both numbers.

The student correctly found LCM(3, 4) = 12, but then multiplied by 6 instead of finding the LCM of 12 and 6. This is the “multiply instead of finding LCM” error applied at the second step. For three numbers, you find LCM(a, b) first, then find LCM(result, c) — not result × c.

LCM(12, 6) = 12, because 12 is already a multiple of 6. So the correct answer is 12, not 72. A quick check: 12 ÷ 3 = 4 ✔, 12 ÷ 4 = 3 ✔, 12 ÷ 6 = 2 ✔. All three divide evenly into 12, and no smaller number achieves this.