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Divisibility by 3 and 9: Add the Digits

Level 2 Divisibility rules · Reviewed by The NumThink editorial team · Last reviewed

Short version: Add the digits. If the sum divides by 3 (or 9), so does the number — and the digit sum leaves the same remainder as the number. For 9, keep adding the digits: 3,462 → 15 → 6, so 3,462 ÷ 9 leaves 6. For 3, divide the sum by 3.

This page is part of the NumThink Divisibility rules course. Read why the method works, watch it run on worked examples, then drill it in the practice box below until it feels automatic.

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Why it works

10 is one more than 9, so every 10, 100 and 1,000 is "a multiple of 9, plus 1": 100 = 99 + 1. A digit 4 in the hundreds is worth 4 × 99 plus 4 — and the 4 × 99 part divides exactly by 9.

Take away all those multiples of 9 and what is left is just the sum of the digits. The same works for 3, because 9 is a multiple of 3.

The steps

  1. Add the digits. 3,462: 3 + 4 + 6 + 2 = 15.
  2. Repeat until one digit is left. 1 + 5 = 6.
  3. Read the answer. ÷9: the remainder is 6 (a final digit sum of 9 would mean remainder 0). ÷3: 6 divides by 3, so 3,462 divides by 3.

Worked examples

Easy Remainder of 2477 ÷ 3

Show the thinking

2 + 4 + 7 + 7 = 20

20 ÷ 3 leaves 2

So 2,477 ÷ 3 leaves 2.

Answer: 2

Medium Remainder of 6147 ÷ 9

Show the thinking

6 + 1 + 4 + 7 = 18

1 + 8 = 9, a multiple of 9

So 6,147 ÷ 9 leaves 0: it divides exactly.

Answer: 0

Hard Remainder of 41838 ÷ 9

Show the thinking

4 + 1 + 8 + 3 + 8 = 24

2 + 4 = 6

So 41,838 ÷ 9 leaves 6.

Answer: 6

When to use it

  • Sharing: 522 cards between 9 players works, because 5 + 2 + 2 = 9 — each gets 58.
  • Checking multiplication ("casting out nines"): for 47 × 23 = 1,081, the digit sums reduce to 2 and 5; 2 × 5 = 10, which reduces to 1 — and 1,081 also reduces to 1 ✓.

When NOT to use it

  • Any other divisor: the digit-sum trick works only for 3 and 9 (because 10 − 1 = 9). For 4, 7 or 11 it gives wrong answers.
  • Small numbers: 27 ÷ 9 is just a times-table fact.

Common mistakes

Using the digit sum for 6. Deciding 15 divides by 6 because its digits add up to 6. Fix: 6 needs both 2 and 3: 15 is odd, so it does not divide by 6 (see the next lesson).
Reading a final digit sum of 9 as remainder 9. Saying 6,147 ÷ 9 leaves 9. Fix: A remainder is always less than the divisor: a digit sum of 9 means remainder 0.
Check yourself: Multiply back: 9 × 683 = 6,147, so the remainder is 0 ✓.