How to tell if a number is divisible by 13
13 works the way 7 does, one place along. Take away 13s you already know until what is left is small enough to recognise, or chop off the last digit, multiply it by 4 and add it back. Neither method asks you to memorise anything new.
Take away multiples of 13 you already know until what is left is small enough to recognise from the 13 times table.
How to check it
- Find a multiple of 13 that fits: 13 × 100 = 1300, 13 × 10 = 130.
- Take it away and look at what is left.
- Repeat until the leftover is inside the 13 times table.
- Landing on 0 means the number divides by 13.
Worked examples
Is 1183 divisible by 13?
1183 − 1170 (13 × 90) = 13 13 = 13 × 1
Nothing left over, so yes. 1183 = 13 × 91.
Is 1147 divisible by 13?
1147 − 1040 (13 × 80) = 107 107 − 104 (13 × 8) = 3
3 left over, so no. 1147 = 13 × 88 remainder 3.
Chop, multiply by 4, add
Chop off the last digit, multiply it by 4, and add it to what is left. Repeat until the number is small enough to judge.
1183 → 118 + 4 × 3 = 130 → 13 + 4 × 0 = 13, which is 13 × 1. So 1183 divides by 13.
It works because 40 is one more than 39, and 39 is three 13s. Multiplying the last digit by 4 and adding it back does not change whether the number divides by 13. It does change what is left over, so this method answers yes or no and never gives the remainder.
This is the same shape as Chika Ofili's rule for 7, where the multiplier is 5 because 50 is one more than 49.
Why these methods work
Taking away 13s cannot change the answer. Every chunk that comes off is a whole number of 13s, so what is left divides by 13 exactly when the number you started with does, and the final leftover is the true remainder.
Neither method is a digit rule, and there is no digit rule to be had. Those come from a place value sitting next to a multiple of the divisor. Ten is one more than 9 (so digits work for 3 and 9); a hundred is one more than 99, which is nine 11s (so pairs work for 11).
For 13, ten leaves 10, a hundred leaves 9 and a thousand leaves 12. Nothing lines up, so no pattern in the digits can decide the question.
Where it goes wrong
- Trying a digit sum. It says nothing about 13.
- Confusing the multiplier with the one for 7. It is 4 for 13 and 5 for 7, and mixing them up gives confident wrong answers.
- Reading the leftover as a factor. A leftover of 3 does not mean the number divides by 3.
- Expecting the chop-and-add result to be the remainder. Only taking away 13s gives you that.
Questions people ask
Why do 7, 11 and 13 come up together?
Because 7 × 11 × 13 = 1001, which is why splitting a long number into three-digit blocks and alternating plus and minus tests all three at once.
Is 13 taught in primary school?
Rarely as a divisibility rule. It shows up in prime factorisation work, where trial division is the honest method.