How do I know whether a word problem needs the HCF or the LCM?

If you are splitting or sharing things into the largest equal groups, you need the HCF. If you are waiting for repeating events to line up or finding a common total, you need the LCM. Ask: am I breaking down (HCF) or building up (LCM)?

Is the HCF the same as the GCF?

Yes. 'Highest common factor' (HCF) and 'greatest common factor' (GCF) are two names for exactly the same thing — the largest number that divides into both values.

Lowest Common Multiple and Highest Common Factor | Math

Two useful number tools

When you work with two or more numbers, two questions come up again and again:

What is the biggest number that divides into both of them? — the Highest Common Factor (HCF).
What is the smallest number that both of them divide into? — the Lowest Common Multiple (LCM).

These two tools solve a surprising range of real problems, from sharing sweets fairly to working out when two events happen at the same time. They build directly on factors and multiples, so keep that lesson handy.

Finding the HCF by listing

The HCF is the largest factor that two numbers share.

Example 1 — Find the HCF of 24 and 36.

List the factors of each number.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
The common factors are 1, 2, 3, 4, 6 and 12. The highest is 12.

So the HCF of 24 and 36 is 12.

Finding the LCM by listing

The LCM is the smallest multiple that two numbers share.

Example 2 — Find the LCM of 6 and 8.

List the multiples of each number.
Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 8: 8, 16, 24, 32...
The smallest number in both lists is 24.

So the LCM of 6 and 8 is 24.

The faster way: prime factors

Listing works but gets slow with big numbers. A quicker method uses prime factorisation.

First write each number as a product of primes:

24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3

	2	2	2	3	3
24	✓	✓	✓	✓
36	✓	✓		✓	✓

For the HCF, multiply the primes they share: 2 × 2 × 3 = 12.
For the LCM, multiply every prime, taking the highest count of each: 2 × 2 × 2 × 3 × 3 = 72.

This method always works, even when the numbers are large. To learn it fully, see prime factorisation.

Choosing HCF or LCM in word problems

The hardest part is deciding which tool a problem needs. Here is the key idea (the why):

HCF problems ask you to break things into the largest equal groups. Words like share, split, cut, equal pieces, greatest are clues.
LCM problems ask when repeating things line up or for a common total. Words like again at the same time, both, smallest amount, next time are clues.

Example 3 (HCF) — Sharing. You have 18 pencils and 24 erasers and want to make identical gift bags with nothing left over. The most bags you can make is the HCF of 18 and 24 = 6 bags.

Example 4 (LCM) — Repeating events. One light flashes every 12 seconds, another every 18 seconds. They flash together at the LCM of 12 and 18 = 36 seconds.

A practice activity

Make a "match the method" game:

Write six short word problems on cards — three about sharing, three about repeating events.
Shuffle them and take turns drawing one.
Decide aloud whether it needs the HCF or the LCM, then solve it.
Challenge: invent your own LCM problem about two buses leaving a station at different intervals, and ask a partner to solve it.

Where this leads

The HCF helps you simplify fractions to their lowest terms, while the LCM helps you find common denominators when adding fractions. Master both methods now — especially the prime-factor approach — and these later topics become far easier.

Lowest Common Multiple and Highest Common Factor

Key takeaways