Maths › Factors, multiples & special numbers
Factors, multiples & special numbers
These are the patterns hiding inside numbers — what divides into what, which numbers are “special”, and the quick tricks for spotting them. Learn a few of these by heart and a lot of questions turn into a few seconds’ work.
What these questions are
Most of this topic is about looking at a number and seeing how it is built — which numbers divide into it, which times-table it belongs to, and whether it is one of the “special” numbers like a square or a prime.
None of it needs hard arithmetic. It rewards knowing a few facts by heart and spotting a pattern quickly. Here is how the page is split:
- Factors, multiples, HCF & LCM — what divides into a number, its times-table, and the shared ones between two numbers.
- Primes, prime factors & divisibility — numbers with exactly two factors, breaking a number into primes, and quick tests for dividing.
- Squares, cubes, roots & triangular numbers — the special number patterns, and powers written the short way.
- Roman numerals, odd & even, and consecutive numbers — a few smaller ideas that come up often.
How to crack them
- Factors divide into a number; multiples are its times-table. Keep the two words apart.
- To spot a prime, test the small primes in turn: 2, then 3, then 5, then 7…
- Learn the divisibility shortcuts — they save you doing the division at all.
- Keep a mental list of the squares to 12² and the cubes to 5³.
A little tip: in every question we’ve bolded the key word — like factor, multiple, prime, highest or lowest — so you always know exactly what you’re being asked to find.
1. Factors, multiples, HCF & LCM
A factor is a number that divides into another exactly, leaving no remainder. A multiple is what you get in a number’s times-table. So 3 is a factor of 12, and 12 is a multiple of 3 — the same fact seen from both ends.
When you compare two numbers, the factors they share are their common factors, and the biggest of those is the highest common factor (HCF). The first multiple they share is the lowest common multiple (LCM).
Listing factors in pairs
The safe way to find every factor is to work in pairs, so you never miss one.
- Work upwards in pairs. 1 goes with 24, 2 with 12, 3 with 8, 4 with 6 — each pair multiplies to make 24.
- Stop when the pairs meet. 5 doesn’t divide 24, and after 4 × 6 the next factor (6) has already appeared, so the list is finished.
- Write them in order, smallest first.
Highest common factor & lowest common multiple
- HCF — list the factors of each, then pick the biggest shared one. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Shared: 1, 2, 3 and 6 — the highest is 6.
- LCM — list the multiples of each, then pick the first shared one. Multiples of 4: 4, 8, 12, 16… Multiples of 6: 6, 12, 18… The first in both lists is 12.
- Keep them straight. The HCF is a factor, so it divides in and stays small; the LCM is a multiple, so it’s usually bigger. Swapping the two is the classic slip.
Your go: factors, multiples, HCF & LCM
Read the bolded word carefully — a factor divides in, a multiple is in the times-table, and watch the highest/lowest swap.
2. Prime numbers, prime factors & divisibility
A prime number has exactly two factors: 1 and itself. So it can only be divided cleanly by 1 and by the number you started with. The first few are 2, 3, 5, 7, 11, 13…
Two things worth remembering: 1 is not prime (it has only one factor, itself), and 2 is the only even prime — every other even number has 2 as a factor.
Is it prime? Test the small primes
- Try the small primes in turn. Is 51 even? No, so 2 is out.
- Test 3 with the digit trick. 5 + 1 = 6, which is a multiple of 3, so 3 divides 51.
- Finish the division. 51 ÷ 3 = 17, so 51 = 3 × 17 — it has more than two factors.
Breaking a number into its prime factors
- Peel off the smallest prime you can, again and again. 60 = 2 × 30, then 30 = 2 × 15, then 15 = 3 × 5.
- Collect the primes. That leaves 60 = 2 × 2 × 3 × 5, all of them prime.
- Check by multiplying back. 2 × 2 × 3 × 5 = 60. In short (index) form that’s 2² × 3 × 5.
Divisibility shortcuts
These quick tests tell you whether one number divides into another, without doing the division:
- ÷ 2 — the number is even (ends in 0, 2, 4, 6 or 8).
- ÷ 3 — the digits add up to a multiple of 3.
- ÷ 4 — the last two digits make a multiple of 4.
- ÷ 5 — it ends in 0 or 5.
- ÷ 6 — it passes the ÷ 2 test and the ÷ 3 test.
- ÷ 9 — the digits add up to a multiple of 9.
- ÷ 10 — it ends in 0.
- Add the digits. 9 + 1 + 8 = 18.
- Check the total. 18 is a multiple of 9, so yes.
- Don’t confuse it with the 3-test. For 9 the digit-total must reach a multiple of 9, not just of 3.
Your go: primes, prime factors & divisibility
Spot the prime, choose the correct prime factorisation, or use a divisibility test.
3. Square & cube numbers, roots & triangular numbers
A square number is a number multiplied by itself: 5² = 5 × 5 = 25. A cube number is a number multiplied by itself three times: 2³ = 2 × 2 × 2 = 8. It’s worth knowing these by heart, so here they are:
A root asks the reverse question. A square root, written √, asks “what number times itself makes this?” A cube root asks “what number times itself three times makes this?”
The small raised number is called an index (or power). It tells you how many of the number to multiply together — it is not a “times”. So 3⁴ means four 3s multiplied, not 3 × 4.
- Square: a number times itself. 7² = 7 × 7 = 49.
- Cube: a number times itself three times. 2³ = 2 × 2 × 2 = 8.
- Index: multiply that many of the number. 3⁴ = 3 × 3 × 3 × 3 = 81. (The trap is 3 × 4 = 12 — the 4 counts the threes, it doesn’t multiply.)
- Root: the reverse. √81 asks what times itself is 81, which is 9. The cube root of 27 asks what times itself three times is 27, which is 3.
Triangular numbers
Triangular numbers are the counts of dots that build up into neat triangles: 1, 3, 6, 10, 15, and so on.
Each triangle adds a new bottom row: +2 dots, then +3, then +4 — the gap grows by one each time.
- Look at the gaps. 1 to 3 is +2, 3 to 6 is +3, 6 to 10 is +4, 10 to 15 is +5.
- The gap grows by one each time. So the next gap must be +6.
- Add it on. 15 + 6 = 21.
Your go: squares, cubes, roots & triangular numbers
Square or cube a number, find a root, evaluate an index, or continue a pattern.
4. Roman numerals, odd & even, and consecutive numbers
Roman numerals build numbers from seven letters. You add the values as you read left to right — unless a smaller letter sits in front of a larger one, which means “subtract” (so IV is 5 − 1 = 4, and IX is 10 − 1 = 9).
Two more small ideas round off the topic. Even numbers end in 0, 2, 4, 6 or 8; odd numbers end in 1, 3, 5, 7 or 9. Adding two odds always gives an even (odd + odd = even), and multiplying by an even number always gives an even answer.
Consecutive numbers are ones that follow straight on, like 7, 8, 9.
Reading a Roman numeral
- Break it into parts. XX is 10 + 10 = 20. Then comes IV.
- Watch the order. I (1) is before V (5), so it means “one less than five” — that’s 4, not 6.
- Add the parts. 20 + 4 = 24.
Consecutive numbers
- Consecutive means one after another. 6, 7 and 8 follow straight on.
- Add them. 6 + 7 + 8 = 21.
- Spot the shortcut. 21 is exactly 3 × 7 — three times the middle number. This works for any three consecutive numbers.
Your go: Roman numerals, odd/even & consecutive
Read or write a Roman numeral, reason about odd and even, or use the consecutive-numbers shortcut.
That’s factors, multiples & special numbers
Learn a handful of these facts by heart — the squares, the cubes, the small primes — and a lot of the 11+ number questions become quick wins. When you’re ready, head back and pick another topic.
Back to Maths