Multiples and divisors of a natural number — Mathematics, 11–13
Divisors and multiples and how they relate: remainder zero, finitely many divisors, endless multiples.
Divisors and multiples: when the division leaves remainder zero
Divide 20 by 4: you get 5 and the remainder is 0. When the remainder of the whole-number division of the larger by the smaller is zero, we say 4 is a divisor of 20 and 20 is a multiple of 4. The multiples of 4 are the numbers in the 4 times table: 4, 8, 12, 16, 20… Every non-zero number is a multiple and a divisor of itself (20 = 20 × 1), and 1 is a divisor of every natural number.
Why a multiple of a multiple is still a multiple
Colour the multiples of 3 in a number table, and the multiples of 9 in another colour. Every multiple of 9 gets both colours. Why? Because 9 = 3 × 3, so 18 = 9 × 2 = 3 × 3 × 2 = 3 × 6. It works for divisors too: 3 divides 12 and 12 divides 60, so 3 divides 60, because 60 = 12 × 5 = 3 × 4 × 5 = 3 × 20. This is not luck with one example: chaining the factors always works.
The divisors of 20, in pairs
Test 1, 2, 3… and see where the remainder is zero. 20 ÷ 1 = 20; 20 ÷ 2 = 10; 20 ÷ 3 = 6 remainder 2 (no); 20 ÷ 4 = 5; 20 ÷ 5 = 4 (we have seen 4 already, so we can stop). Each divisor brings a partner: 1 and 20, 2 and 10, 4 and 5. The set of divisors of 20 is {1, 2, 4, 5, 10, 20}: 4 ∈ the set, 3 ∉. There are six and no more. The multiples of 20 — 20, 40, 60, 80… — never run out.
The trap: mixing up divisor and multiple
A common slip: saying 5 is a multiple of 20 because ‘they go together’. It is not: 20 is a multiple of 5 (20 = 5 × 4) and 5 is a divisor of 20. Rule of thumb: a divisor of a number is never bigger than it; a non-zero multiple is never smaller. That is why a number has few divisors (20 has six) but infinitely many multiples. And do not forget to list 1 and the number itself among the divisors.
Where you see this outside school
When you share 12 biscuits among friends with none left over, you are looking for divisors of 12: 1, 2, 3, 4, 6 or 12 friends. Arranging 24 chairs in equal rows is the same. Multiples show up in anything that repeats: if a bus comes every 15 minutes, it comes at 15, 30, 45, 60… minutes, the multiples of 15. Whenever something repeats at equal steps, you are counting multiples.
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