Factors & Multiples Worksheets

Four types, one idea

Factors asks you to list everything that divides a number. Multiples asks for its first six. HCF and LCM take a pair of numbers. Mixed balances all four across the sheet.

Sixteen questions by default, numbers up to 60. Both adjustable, answer key included.

Factors go in, multiples go out

The two words get confused for years, and the fix is one sentence rather than a definition.

Factors are smaller than the number and there are finitely many. Multiples are bigger and there are infinitely many. A pupil who has that straight will not write "the factors of 6 are 6, 12, 18" again, which is far and away the commonest error on this topic.

List factors in pairs

The method that stops pupils missing one. For 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Then stop, because 5 does not divide and the next pair after that would repeat.

Working in pairs from the outside inwards is systematic and it makes it obvious when to stop. It also quietly demonstrates that a square number has an odd count of factors, the middle pair being one number twice over. Nicer discovered than told.

HCF and LCM

Listing everything out works for small numbers, and it stops working quickly.

Prime factorisation is the method that scales: for the HCF take the primes common to both, at the lowest power; for the LCM take every prime that appears, at the highest. Teach it as soon as the listing method starts producing errors, which is usually somewhere around 36 and 48.

Here is a useful check. HCF × LCM equals the product of the two numbers, every time, so a pupil who has found both can verify both in a single multiplication.

Where it is actually used

Simplifying fractions is the HCF. Adding fractions is the LCM — the common denominator is exactly that, and pupils who find fraction addition mysterious often just have not connected the two ideas.

Beyond fractions: sharing into equal groups, working out when two repeating events coincide, and any question involving tiles or packs or arrangements. The classic buses-leaving-the-station question is an LCM in a coat.

Setting the ceiling

Sixty is the default and it is well chosen — numbers up to 60 have interesting factor structures without the arithmetic becoming the obstacle. Raise it to 100 once the method is secure; go higher only if prime factorisation is being taught alongside.

Related: prime numbers, which is where the reliable method for both comes from.