LCM and HCF Worksheets

Three modes

Rows of number pairs. You choose what gets asked — the highest common factor, the lowest common multiple, or both of them for every pair on the sheet.

Set how many questions and how large the numbers go, with an answer key if you want one.

Which is which

The names get confused constantly, and the fix is to notice that each one tells you where to look.

The highest common factor is a factor, so it is smaller than both numbers — the biggest thing that divides into both. The lowest common multiple is a multiple, so it is at least as big as the larger number — the smallest thing they both divide into.

A pupil who answers 48 for the HCF of 12 and 16 has not made an arithmetic error. They have solved the wrong problem, and telling them to check their working will not help.

Listing works, then stops working

For small pairs, write out the factors of each and take the biggest appearing in both. Perfectly sound. It collapses somewhere around 36 and 48.

Prime factorisation is what scales. Break both numbers down, then for the HCF take the primes common to both at the lowest power, and for the LCM take every prime that appears at the highest power. Same decomposition, two different readings of it.

One check catches everything

HCF × LCM equals the product of the two numbers. Always.

For 12 and 18: HCF 6, LCM 36, and 6 × 36 = 216 = 12 × 18. A pupil who has found both can verify both in a single multiplication, which turns this from a topic with no feedback into one where they know immediately whether they are right.

Where it is actually needed

Fractions, mostly. Pupils rarely make the connection unprompted.

Simplifying a fraction means dividing top and bottom by their HCF. Adding fractions means finding a common denominator, which is the LCM. A child who finds fraction arithmetic mysterious very often just has not been told that these are the same two operations under different names.

Beyond that: sharing into equal groups, working out when two repeating events coincide, and every buses-leaving-the-station question ever set.

Setting the ceiling

Keep it modest while the method is being learnt. The reasoning is the point, and large numbers simply move the obstacle to the arithmetic.

Related: prime numbers for the factorisation this depends on, and factors and multiples.