Question 1
The two numbers are 16 and 28.
What is the least common multiple of 16 and 28?
- 111
- 113
- 16
- 224
- 112
Show the worked solution
- GCD(16, 28) = 4.
- LCM = 16 × 28 ÷ 4 = 112.
- Answer: 112.
GCF & LCM
GCF and LCM problems — one identity does both jobs
GCF and LCM problems are solved fastest through a single identity: for any two positive integers, the GCF multiplied by the LCM equals the product of the numbers. Find the GCF by inspection or the Euclidean algorithm, then divide the product by it. Factorising both numbers fully is almost never necessary.
Most candidates learn these through prime factorisation, which is reliable and slow. Under a clock the identity is far better: the greatest common factor of two exam-sized numbers is usually obvious within a couple of seconds, and everything else is one division. The factor-tree habit is what costs the time here, not the difficulty.
Try it against the clock. HCF & LCM runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.
How to solve GCF and LCM problems with the product identity
-
Find the GCF first, by inspection
For 16 and 24 the largest shared factor is 8. Exam pairs are built so this is visible immediately; only reach for the Euclidean algorithm when it is not.
-
Apply GCF times LCM equals the product
With 16 and 24 the product is 384, so the LCM is 384 divided by 8, which is 48. One multiplication and one division replace two factor trees.
-
Divide before you multiply
Compute 16 divided by 8 to get 2, then 2 times 24 to get 48. Cancelling first keeps the numbers small and avoids a three-digit multiplication under time pressure.
-
Check the coprime case separately
When the GCF is 1, as with 8 and 35, the LCM is simply the product: 280. Spotting coprimality first turns the whole question into a single multiplication.
Reading the pair before you compute
| The two numbers | What that tells you |
|---|---|
| Share no common factor | GCF is 1, LCM is the product |
| One divides the other | GCF is the smaller, LCM is the larger |
| Both even | GCF is at least 2 — halve and recheck |
| 16 and 24 | GCF 8, so LCM is 384 ÷ 8 = 48 |
| 8 and 35 | Coprime, so LCM is 280 |
3 sample gcf and lcm problems, with worked solutions
Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
Question 2
The two numbers are 9 and 24.
What is the least common multiple of 9 and 24?
- 72
- 71
- 9
- 33
- 24
Show the worked solution
- GCD(9, 24) = 3.
- LCM = 9 × 24 ÷ 3 = 72.
- Answer: 72.
Question 3
The two numbers are 10 and 18.
What is the least common multiple of 10 and 18?
- 10
- 18
- 89
- 91
- 90
Show the worked solution
- GCD(10, 18) = 2.
- LCM = 10 × 18 ÷ 2 = 90.
- Answer: 90.
Common mistakes this drill is built from
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
Building full factor trees every time
Prime factorisation always works and always costs more seconds than the identity. Reserve it for awkward three-digit pairs.
Swapping the GCF and the LCM
The GCF is never larger than the smaller number and the LCM is never smaller than the larger one. A quick size check catches the swap instantly.
Multiplying before cancelling
Forming the full product first creates a larger division than necessary. Cancel the GCF out of one number, then multiply.
Why your answer came out wrong
The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.
Is the LCM of 8 and 35 equal to 40?
No — it is 280. 8 and 35 share no common factor, so the GCF is 1 and the LCM is the full product. 40 is what you get by assuming a shared factor of 7 that only one of the numbers has. Check coprimality before reaching for anything clever.
I computed 384 and picked it as the LCM of 16 and 24. What went wrong?
384 is the product of the two numbers, not the LCM. The identity says the product equals the GCF times the LCM, so you still have to divide by the GCF of 8 to reach 48. Stopping at the product is the most common slip with this method.
FAQ
What is the quickest way to find the LCM of two numbers?
Find the greatest common factor, then divide the product of the two numbers by it. For 16 and 24 the GCF is 8 and the product is 384, giving an LCM of 48. Cancel the GCF out of one number before multiplying to keep the arithmetic small.
Does the GCF times LCM identity always hold?
For any two positive integers, yes — it is exact, not an approximation. It does not extend to three or more numbers, where the product of the GCF and the LCM is generally not the product of all of them. For pairs, which is what these questions ask, it is completely reliable.
How do I find the GCF when it is not obvious?
Use the Euclidean algorithm: divide the larger by the smaller, then replace the larger with the remainder and repeat until the remainder is zero. The last non-zero remainder is the GCF. It takes two or three short steps even for awkward pairs.
Practise the next topic
- All 51 drill types — the shared daily cycles 36 of 51, with every published time limit.
- Number properties questions — count the terms — number properties questions, timed.
- Simplification questions — BODMAS — simplification questions, timed.
- Speed tricks — the seven mental-math shortcuts that make a 15-second answer possible.
- How to improve calculation speed — the six-week arc these drills fit into.
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