Divisibility by 9

The idea

Each place value is 1 more than a multiple of 9, so a number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Not just the units digit

For 10, 5 and 2 the units digit told us everything. Now try 9. The numbers 99 and 109 both end in the digit 9, and 99 = 9 × 11 is divisible by 9.

Is 109 divisible by 9 too?

• Yes, it ends in 9 like 99
• No
• Only some numbers ending in 9 are

Peel off the nines

Every place value is 1 more than a multiple of 9. Each digit peels off a multiple of 9 (blue) and leaves one cell per unit of the digit (yellow). Change digits with ▲ ▼.

The digit sum decides

Each place value is 1 more than a multiple of 9, so a number is divisible by 9 if and only if the sum of its digits is divisible by 9.

The place values: 1 = 0 + 1 10 = 9 + 1 100 = 99 + 1 1000 = 999 + 1 and so on.

For 7309: 7309 = (7 × 999 + 3 × 99 + 0 × 9) + (7 + 3 + 0 + 9)

The first bracket is a multiple of 9. So 7309 is divisible by 9 exactly when 7 + 3 + 0 + 9 = 19 is. It is not.

A number made only of the digits 0 and 9, such as 99009, is always divisible by 9. Every digit is 0 or 9, so the digit sum is a multiple of 9.

Notes

Each place value is 1 more than a multiple of 9, so a number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Check yourself

Which of these numbers is divisible by 9?

The number 47★2 has a missing digit ★. Which digit makes it divisible by 9?

Answer: 5

4 + 7 + 2 = 13. The next multiple of 9 above 13 is 18, and 18 − 13 = 5. So ★ = 5: 4752 has digit sum 18 and 4752 = 9 × 528.

Is 99009 divisible by 9?

4086 is divisible by 9. Is 8460, made from the same digits in a different order, divisible by 9 too?

• 6209. It ends in 9, but 6 + 2 + 0 + 9 = 17 is not divisible by 9.
• 3410. The digits add up to 3 + 4 + 1 + 0 = 8, which is not divisible by 9.
• 2475 — correct. Yes! 2 + 4 + 7 + 5 = 18 = 9 × 2, so 2475 is divisible by 9.
• 9001. It starts with 9, but 9 + 0 + 0 + 1 = 10 is not divisible by 9.
• Yes, because 9 + 9 + 0 + 0 + 9 = 27, a multiple of 9 — correct. Yes! 27 = 9 × 3. Every digit is 0 or 9, so each term of the expanded form is a multiple of 9.
• No, because it is too big to check. Size does not matter. The digit sum 27 is quick to check.
• No, because it has zeros in it. Zeros add nothing to the digit sum. The digit sum is 27, a multiple of 9.
• No, because the units digit changed. For 9 the units digit alone does not decide. Look at the digit sum.
• Yes, because the digit sum is the same, 18 — correct. Yes! 8 + 4 + 6 + 0 = 18, just like 4 + 0 + 8 + 6. Order does not change the digit sum.
• Only if the thousands digit stays the same. Nothing about one place matters. Only the total of the digits does.