Tests for 3 and 9

Three friends win 471 sweets in a competition and want to share them equally. Will it come out fair, with none left over? The last digit is 1, but in the Tests for 2, 5 and 10 lesson we saw that the last digit only works for 2, 5 and 10. For 3 we need a new trick.

Add up the digits

Here is the test. Add up the digits of the number: this is called the digit sum.

471: the digit sum is 4 + 7 + 1 = 12.
12 divides by 3 (12 ÷ 3 = 4).
So 471 divides by 3 too. Each friend gets 157 sweets.

The test for 3: a number divides by 3 if its digit sum divides by 3. The test for 9 works the same way: a number divides by 9 if its digit sum divides by 9. For example, 3,456 has digit sum 3 + 4 + 5 + 6 = 18, and 18 divides by 9, so 3,456 does too.

If the digit sum is still big, add up its digits again: 99,999 has digit sum 45, and 45 has digit sum 9. So 99,999 divides by 9.

Why it works: a ten is a nine plus one

This test looks like magic, but it has a reason. Take one ten. Put 9 of its dots aside: 9 splits into threes (three threes) and into nines. One dot is left over. So every ten is a nine plus one left over.

A row of ten dots. Nine are blue and grouped into three threes; the tenth dot is orange and sits apart: every ten is a nine plus one

Now take 42: 4 tens and 2 ones.

Each of the 4 tens leaves 1 over after its nine. That is 4 left over.
The 2 ones are left over too.
So all that is left over is 4 + 2 = 6 - the digit sum!

The nines are no trouble: they always split into threes and nines. Only the leftovers decide, and the leftovers are the digit sum. 6 divides by 3, so 42 divides by 3 (42 ÷ 3 = 14). 6 doesn't divide by 9, so 42 doesn't divide by 9: in fact it leaves exactly 6 over (4 nines are 36, and 36 + 6 = 42).

The number 42 drawn as four rows of ten dots and two loose dots. In each row nine dots are blue and one is orange. The four orange dots and the two loose dots are gathered together: six dots left over

The same is true of hundreds: 100 is 99 + 1, and 99 is eleven nines. So every hundred also leaves exactly 1 over, and every digit counts its own leftovers.

A test that fails

Be careful not to mix the tests up. For 3 and 9 the last digit tells you nothing: 13 ends in 3 but doesn't divide by 3, and 12 ends in 2 but does. For 2, 5 and 10 the digit sum tells you nothing: 14 and 41 have the same digit sum, but only 14 is even. Each test works only for the reason behind it.

Try it at home

Pick any number on a receipt or a car number plate. Find its digit sum. Does the number divide by 3? By 9?
Divisible by 3, by 9, or neither: 111; 2,025; 1,000?
A number divides by 9. Must it divide by 3? Answers below.

Answers: 111 by 3 only (digit sum 3); 2,025 by both (digit sum 9); 1,000 by neither (digit sum 1). Yes: a nine is three threes, so anything that splits into nines also splits into threes.

Words to know

digit sum - what you get by adding up the digits of a number: the digit sum of 471 is 4 + 7 + 1 = 12
test for 3 (or 9) - a number divides by 3 (or 9) if its digit sum does

Big idea

Every ten is a nine plus one, and every hundred is 99 plus one. The nines always split into threes and nines, so only the leftovers decide - one for each ten, one for each hundred, plus the ones - and the leftovers add up to the digit sum.