How to tell if a number is divisible by 7
This is long division said out loud: 7 × 600 is 4200, take it away, look at what is left. There is nothing to memorise, and it works on any number, however long.
Take away multiples of 7 you already know until what is left is small enough to recognise from the 7 times table.
How to check it
- Find a multiple of 7 you know that fits inside the number: 7 × 100 = 700, 7 × 600 = 4200, and so on.
- Take it away and look at what is left.
- Repeat until the leftover is small enough to recognise from the 7 times table.
- If you land on 0, the number divides by 7. Anything else is the remainder.
Worked examples
Is 4620 divisible by 7?
4620 − 4200 (that is 7 × 600) = 420 420 = 7 × 60
Nothing left over, so yes. 4620 = 7 × 660.
Is 1147 divisible by 7?
1147 − 700 (7 × 100) = 447 447 − 420 (7 × 60) = 27 27 − 21 (7 × 3) = 6
6 left over, so no. 1147 = 7 × 163 remainder 6.
The rule you will meet at school: double and subtract
This is the one most textbooks print: chop off the last digit, double it, and subtract it from what is left. Repeat until the number is small enough to recognise.
4620 → 462 − 0 × 2 = 462 → 46 − 2 × 2 = 42, and 42 is 7 × 6. So 4620 divides by 7.
It works because 20 is one less than 21, and 21 is three 7s. Doubling the last digit and taking it away does not change whether the number divides by 7 — the same reasoning as Chika's trick below, with the multiplier going the other way.
The catch is that it can ask you to take a bigger number from a smaller one. 98 chops to 9 − 16 = −7, which is a correct answer because −7 is a multiple of 7, but a negative number in the middle of a shortcut is where most people lose their place. That is the main reason to prefer one of the other two.
Chika's trick: chop, multiply by 5, add
In 2019 a twelve-year-old called Chika Ofili was given a holiday maths book and came back with a genuinely new shortcut for 7: chop off the last digit, multiply it by 5, and add it to what is left. Repeat until the number is small enough to judge.
4620 → 462 + 0 × 5 = 462 → 46 + 2 × 5 = 56, and 56 is 7 × 8. So 4620 divides by 7.
It works because 50 is one more than 49, and 49 is seven 7s. Multiplying the last digit by 5 and adding it back does not change whether the number divides by 7. It does change what is left over, so the trick answers yes or no and never gives the remainder: 15 becomes 26, and 26 leaves 5 over where 15 left 1.
Adding never goes negative, which is the practical advantage over doubling and subtracting.
Why these methods work
Taking away 7s cannot change the answer. Every chunk that comes off is a whole number of 7s, so what is left divides by 7 exactly when the number you started with does, and the final leftover is the true remainder.
The two chopping tricks are the same idea with different multipliers. Multiplying the last digit by m and adding turns the number N into N + (10m − 1) × the digit, so any m where 10m − 1 is a multiple of 7 leaves the answer alone. For 7 that is m = 5, giving 49, and m = −2, giving −21 — which is subtraction. There are no others.
None of the three is a digit rule, because a digit rule needs a place value landing next to a multiple of the divisor. 10, 100 and 1000 leave 3, 2 and 6 when you share them into 7s, so nothing lines up for the digits to catch.
Where mistakes happen
- Stopping a chopping trick too early. Keep going until the result is inside the 7 times table you know.
- Giving up when double-and-subtract goes negative. −7, −14 and −21 are multiples of 7 just as 7, 14 and 21 are, so a minus sign means the answer is yes, not that you went wrong.
- Not being sure of the 7 times table. The method ends by recognising the leftover, so if 56 does not look like 7 × 8, keep taking 7s away one at a time until what is left is smaller than 7.
Questions people ask
Is there really no simple rule for 7?
Not one in the same class as 2, 5 or 10. Every published rule for 7 is an iterative procedure, which is why most curricula do not teach one at primary level.
Which method should I use?
Taking away 7s you already know. It uses times tables you have, it never goes negative, and it gives you the remainder as well as the yes or no. Of the two chopping tricks, Chika's is the kinder one, because adding never takes you below zero.
What is the double-and-subtract rule called?
English has no settled name for it — textbooks simply call it the divisibility rule for 7. Vedic mathematics does name the multipliers: the 2 you subtract is the negative osculator of 7, and Chika's 5 is the positive one.
Does Chika's trick work for other numbers?
For some. It needs a multiple of ten sitting right next to a multiple of the number you are testing, which is exactly what 7 has: 50 is one more than 49.