Simple Divisibility Rules

We all know them… or many of them. But they are great fun (and useful)

2 – if the number ends in 0,2,4,6, or 8, it is divisible by 2
3 – if the sum of the number’s digits is divisible by 3, then the number is divisible by 3
4 – if the number’s last two digits are divisible by 4, then the number is divisible by 4 (if it is not divisible by 2, no need to check – it won’t be divisible by 4)
5 – if the number ends in 0 or 5, the number is divisible by 5
6 – if the number passes the test for 2 and 3 (ends in 0, 2, 4, 6, 8 or 0 and the sum of the digits is a multiple of 3), then the number is divisible by 6
7 – tricky, two methods, future post!
8 – tricky, future post!
9 – if the sum of the number’s digits is divisible by 9, then the number is divisible by 9 (if it is not divisible by 3, no need to check – it won’t be divisible by 9)
10 – if the number ends in 0, it is divisible by 10 (alternately, if it passes the tests for 2 and for 5, then it is divisible by 10)
11 – tricky, with two methods, future post!
13 – tricky, future post!
16 – tricky, future post!
and more and more for future posts.

Some examples:

78 – ends in 2, so divisible by 2. The sum of the digits (7+8=15) is divisible by 3, so 78 is divisible by 3. The last two digits… hm, the number is the last two digits! so we just divide to find that 4 does not go in evenly. The last digit is not 0 or 5, so it is not a multiple of 5. It passes the tests for 2 and 3, so it is a multiple of 6. The sum of its digits is not a multiple of 9, so 78 is not a multiple of 9. And it doesn’t end in 0, so it is not a multiple of 10.

1234 – ends in 2, so is divisible by 2. The sum of the digits (1+2+3+4=10) is not divisible by 3, so 1234 is not divisible by 3. The last two digits, 34, are not divisible by 4, so 1234 is not divisible by 4. It doesn’t end with 0 or 5, so not divisible by 0 or 5. It passes the test for 2, but not for 3, so it is not divisible by 6. The sum of the digits is not divisible by 9, so not divisible by 9. Doesn’t end in 0, not divisible by 10.

271845 – ends in 5, so not divisible by 2. The sum of the digits (2+7+1+8+4+5=27) is divisible by 3, so 271845 is divisible by 3.  It is not divisible by 2, so it won’t be divisible by 4. It ends in a 5, so it is divisible by 5. It passes the test for 3, but fails for 2, so it is not a multiple of 6. The sum of the digits (27) is divisible by 9, so it is divisible by 9. It does not end in 0, so it is not divisible by 10.