Highest Common Factor and Lowest Common Multiple
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
The factors of 12 are: 1, 2, 3, 4, 6, 12. The factors of 18 are: 1, 2, 3, 6, 9, 18. The common factors — numbers that appear in both lists — are 1, 2, 3, and 6. The highest of these is 6.
Alternatively, using prime factorisation: \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\). The HCF takes the lowest power of each shared prime: \(2^1 \times 3^1 = 6\).
Factors of 16: 1, 2, 4, 8, 16. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 4, and 8. Yes, 4 is a common factor — but the HCF is the highest common factor, and 8 is higher. So HCF(16, 24) = 8.
A common mistake is to spot a shared factor like 4 and stop looking. Using prime factorisation confirms: \(16 = 2^4\) and \(24 = 2^3 \times 3\). Taking the lowest power of the shared prime gives \(2^3 = 8\). The HCF isn’t just any common factor — it’s the greatest one.
Take 8 and 9. Factors of 8: 1, 2, 4, 8. Factors of 9: 1, 3, 9. The only factor they share is 1, so HCF(8, 9) = 1. Numbers like this are called coprime.
It doesn’t require prime numbers — any pair of numbers that share no prime factors will have an HCF of 1. For example, 16 and 25 are both composite, but \(16 = 2^4\) and \(25 = 5^2\) share no primes, so their HCF is 1.
12 is already a multiple of 4 (since \(4 \times 3 = 12\)), and 12 is obviously a multiple of itself. So 12 is a common multiple of 4 and 12. Can we find a lower common multiple? Any common multiple must be a multiple of 12, and the first multiple of 12 is 12 itself. So 12 is the lowest.
This illustrates a general rule: when one number is a factor of the other, the LCM is simply the larger number. Students often expect the LCM to be bigger than both numbers, but that isn’t always the case.
Give an Example Of…
For each prompt, provide: an example, another example, one no-one else will think of, and one someone might think works but doesn’t.
Example: 18 and 30
Another: 6 and 48
Creative: 42 and 54 — both are multiples of 6, and HCF(42, 54) = 6 since \(42 = 2 \times 3 \times 7\) and \(54 = 2 \times 3^3\). Or simply 6 and 6 — the HCF of any number with itself is that number!
Trap: 6 and 10 — a student might think “6 is one of the numbers, so 6 must be the HCF.” But 6 is not a factor of 10, so HCF(6, 10) = 2, not 6. For 6 to be the HCF, it must be a factor of both numbers.
Example: 3 and 5 — LCM = 15 = 3 × 5
Another: 4 and 7 — LCM = 28 = 4 × 7
Creative: 1 and 99 — LCM = 99 = 1 × 99. Any pair involving 1 works. Or 11 and 13 — two primes always have an LCM equal to their product.
Trap: 4 and 6 — a student might assume the LCM is always the product, giving \(4 \times 6 = 24\). But LCM(4, 6) = 12, not 24. The rule only works when the two numbers are coprime (HCF = 1). Since 4 and 6 share a factor of 2, their LCM is less than the product.
Example: 3 and 12 — HCF = 3
Another: 5 and 20 — HCF = 5
Creative: 7 and 7 — the HCF of any number with itself is that number! Or 1 and 50 — HCF = 1, which equals the smaller number.
Trap: 6 and 9 — a student might think “6 is a factor of 9” or “6 divides into 9 somehow.” But 6 does not divide evenly into 9, so HCF(6, 9) = 3, not 6. For the HCF to equal one of the numbers, that number must be a factor of the other.
Example: 4 and 6 — HCF = 2, LCM = 12. Check: \(2 \times 12 = 24 = 4 \times 6\) ✓
Another: 10 and 15 — HCF = 5, LCM = 30. Check: \(5 \times 30 = 150 = 10 \times 15\) ✓
Creative: This is always true for any pair of positive integers! So even 1 and 1 works: HCF = 1, LCM = 1, and \(1 \times 1 = 1 \times 1\). The real discovery here is that \(\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b\) is a universal rule.
Trap: Trying it with three numbers, e.g. 2, 3, and 4. HCF = 1, LCM = 12, but \(1 \times 12 = 12 \neq 2 \times 3 \times 4 = 24\). The relationship only holds for pairs of numbers, not triples.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
If a number \(d\) divides both \(n\) and \(n+1\), then it must also divide their difference: \((n+1) – n = 1\). The only positive number that divides 1 is 1 itself. So consecutive numbers can never share a factor greater than 1.
Students can verify: HCF(7, 8) = 1, HCF(20, 21) = 1, HCF(99, 100) = 1. No matter how large the numbers get, consecutive numbers are always coprime.
True case: HCF(6, 10) = 2, and 2 is prime. HCF(15, 25) = 5, and 5 is prime. This happens when the two numbers share exactly one prime factor (possibly raised to different powers).
False case: HCF(8, 12) = 4, which is not prime. HCF(12, 18) = 6, which is not prime. Whenever the two numbers share more than one prime factor (or the same prime more than once), the HCF can be composite. And HCF(7, 10) = 1, which is not prime either.
True case: LCM(3, 5) = 15, which is greater than both 3 and 5. This happens whenever neither number is a multiple of the other.
False case: LCM(3, 9) = 9, which equals 9 but is not greater than it. Whenever one number is a factor of the other, the LCM equals the larger number — so it isn’t greater than both. Also consider LCM(6, 6) = 6, which equals both numbers.
The LCM is a common multiple of both numbers, so it must be at least as large as each of them. The smallest multiple of any positive number \(n\) is \(1 \times n = n\) itself, so the LCM can never drop below either number.
The LCM can equal one of the numbers (e.g. LCM(3, 9) = 9) or exceed both (e.g. LCM(4, 6) = 12), but it can never be smaller than both. This pairs neatly with the previous question: the LCM is sometimes greater than both, but never smaller than both.
Odd One Out
Which is the odd one out? The challenge: make a valid mathematical case for each one being the odd one out.
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: HCF = 24
Reasoning: “I listed the multiples. 12, 24, 36, 48… and 8, 16, 24, 32… The first one they share is 24. So the HCF is 24.”
The student has confused HCF with LCM. They listed multiples instead of factors and found the lowest common multiple (24) rather than the highest common factor. This is one of the most common mix-ups with HCF and LCM.
The correct approach: list the factors. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 8: 1, 2, 4, 8. Common factors: 1, 2, 4. The highest is 4. A useful check: the HCF must be a factor of both numbers, so it can never be larger than either number.
Answer: LCM = 21 ✓
Reasoning: “To find the LCM, you always just multiply the two numbers together. 3 × 7 = 21.”
The answer is correct — LCM(3, 7) is 21 — but the reasoning is dangerously wrong. The student has invented a rule: “the LCM is always the product of the two numbers.” This only works when the numbers are coprime (share no common factors other than 1).
A counterexample breaks the rule immediately: LCM(4, 6). Using the student’s method: \(4 \times 6 = 24\). But the actual LCM is 12, since 12 is the first number in both the 4 and 6 times tables. The product gives a common multiple, but not necessarily the lowest.
Answer: HCF = 72
Reasoning: “24 = 2³ × 3 and 36 = 2² × 3². For the HCF, I take the highest power of each prime: 2³ × 3² = 8 × 9 = 72.”
The student has confused the HCF method with the LCM method. They took the highest power of each prime factor — that’s the rule for finding the LCM, not the HCF. (Indeed, 72 is actually the LCM of 24 and 36!)
For the HCF, you take the lowest power of each shared prime. Both numbers share the primes 2 and 3. The lowest power of 2 is \(2^2\) and the lowest power of 3 is \(3^1\). So HCF = \(2^2 \times 3 = 12\). A quick sense-check confirms: 12 is a factor of both 24 and 36 ✓, and the student’s answer of 72 is larger than both numbers, which is impossible for an HCF.
Answer: LCM = 30
Reasoning: “Multiples of 5: 5, 10, 15, 20, 25, 30… Multiples of 15: 15, 30, 45… They share 15 and 30, but 15 can’t be the LCM because the LCM should be bigger than both numbers. So I picked the next one: 30.”
The student correctly identified 15 as a common multiple but then rejected it based on a false belief: “the LCM must be bigger than both numbers.” This is a common misconception. The LCM is the lowest common multiple — and 15 is the lowest number that appears in both lists.
15 is a multiple of 5 (since \(5 \times 3 = 15\)) and 15 is a multiple of itself. Since 5 is a factor of 15, the LCM is simply 15. There’s no rule saying the LCM has to exceed both numbers — when one number divides the other, the LCM equals the larger one.