Division Tricks
Divisibility by 7 – Double, Subtract, Repeat
Seven is the awkward one, but this double-and-subtract routine tames it.
This page explains the rule, shows why it works, and walks through worked examples. Then you can try it yourself and check your understanding with practice questions. Use the previous and next links at the bottom to move through the tricks in order.
The rule
The Rule: Double the last digit, subtract it from the rest of the number, and if the result is divisible by 7, so is the original number.
Why it works
Why It Works: This one is genuinely less obvious than the others — and that is exactly why it feels clever rather than routine. Splitting off the last digit, doubling it, and subtracting is a way of quietly “cancelling out” the effect that the number 10 has, in a way that happens to line up perfectly with 7. It is not something you can see at a glance; it only falls out once you follow the arithmetic all the way through. But it always works, every single time.
Formal Proof (for advanced learners)
Let N = 10a + b, where b is the units digit and a is the rest of the number. Claim: 7 | N ⇔ 7 | (a − 2b). Multiply (a − 2b) by 10: 10a − 20b. Since −20 ≡ 1 (mod 7) (because −21 = −3·7), we get −20b ≡ b (mod 7), so 10(a − 2b) ≡ 10a + b = N (mod 7). Because gcd(10, 7) = 1, multiplying by 10 never changes whether a number is ≡ 0 (mod 7), so N ≡ 0 (mod 7) ⇔ a − 2b ≡ 0 (mod 7).
Examples
- 203 → double 3 to get 6, then 20 − 6 = 14, which is divisible by 7.
- 168 → double 8 to get 16, then 16 − 16 = 0, which counts as divisible by 7.
- 92 → double 2 to get 4, then 9 − 4 = 5, not divisible by 7.
Step by step
Example A: is 4368 divisible by 7?
- Double the last digit of 4368 (8 → 16) and subtract it from the rest (436): 436 − 16 = 420.
- Double the last digit of 420 (0 → 0) and subtract it from the rest (42): 42 − 0 = 42.
- 42 is a multiple of 7.
- So 4368 is divisible by 7.
Example B: is 9115 divisible by 7?
- Double the last digit of 9115 (5 → 10) and subtract it from the rest (911): 911 − 10 = 901.
- Double the last digit of 901 (1 → 2) and subtract it from the rest (90): 90 − 2 = 88.
- 88 is not a multiple of 7.
- So 9115 is not divisible by 7.
Try it yourself
Now it is your turn. A new random question appears each time. Choose True or False. If you are right, we show how the trick got you there, step by step. If you are not, you will see the full working two ways, the shortcut and the usual method, so you can spot exactly where your answer differed. Press New Question as often as you like: a few quick rounds is the best way to make the trick feel automatic.
Practice questions
Have a go on paper first, then tap Show answer to check your method.
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Is 63 divisible by 7?
Show answer
Yes. Double the last digit of 63 (3 → 6) and subtract it from the rest (6): 6 − 6 = 0. 0 is a multiple of 7. So 63 is divisible by 7.
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Is 85 divisible by 7?
Show answer
No. Double the last digit of 85 (5 → 10) and subtract it from the rest (8): 8 − 10 = −2. −2 is not a multiple of 7. So 85 is not divisible by 7.
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Is 154 divisible by 7?
Show answer
Yes. Double the last digit of 154 (4 → 8) and subtract it from the rest (15): 15 − 8 = 7. 7 is a multiple of 7. So 154 is divisible by 7.
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Is 2003 divisible by 7?
Show answer
No. Double the last digit of 2003 (3 → 6) and subtract it from the rest (200): 200 − 6 = 194. Double the last digit of 194 (4 → 8) and subtract it from the rest (19): 19 − 8 = 11. 11 is not a multiple of 7. So 2003 is not divisible by 7.
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Can 2548 apples be packed equally into 7 baskets with none left over?
Show answer
Yes. Double the last digit of 2548 (8 → 16) and subtract it from the rest (254): 254 − 16 = 238. Double the last digit of 238 (8 → 16) and subtract it from the rest (23): 23 − 16 = 7. 7 is a multiple of 7. So 2548 is divisible by 7.
Tips and watch-outs
- A negative result is fine: 14 → 1 − 8 = −7, still a multiple of 7.
- On long numbers, repeat the routine until you reach something you know: 2548 → 238 → 7, so 2548 is divisible by 7.