Skip this if you don’t like math.

I’ll start with the easy rules. *In base ten, a number is divisible by ten if it ends in zero.* **In base N, a number is divisible by N if it ends in zero.** Thus 45360(base 7) is divisible by 7.

(In bases higher than ten, use A for 10, B for 11, etc.)

*In base ten, if a number is divisible by 5, it ends in a zero or five.* I**n base N if a number is divisible by a factor of N, it ends in a digit that is divisible by that factor.** Thus the following numbers, all in base 15, are divisible by five:

3A, B0, and 95

Also in base 15, the following numbers are divisible by three:

73, 59, 3C

Note: (the even last digit) = (divisible by two) only works for N being an even number.

*In base ten, if the sum of the digits is divisible by three or nine, the number is divisible by three or nine, respectively*. **In base N, if the sum of the digits is divisible by a factor of N — 1, the number is divisible by that factor.**

This example will be from base 13. We look at factors of 13 — 1 = 12

22 is divisible by 2, but not by 4, 6, or 12. In base 13, 22 is represented as 19. 9 + 1 = 10, which is divisible by 2, but not 4, 6, or 12.

102 base 10 is divisible by 6, but not by 4. In base 13, 102 is represented by 7B. 7 + B = 7 + 11 = 18, which is divisible by 6, but not 4.

24 base 10 is divisible by 12. In base 13, 24 is represented by 1B. 1 + B = 1 + 11 = 12, which is obviously divisible by 12.

*In base 10, if alternate digits are added and subtracted, and the result is divisible by 11, then the number is divisible by 11.* For example, using 869 base 10, 8 — 6 + 9 = 11, which is divisible by 11. (Most small numbers add to zero if they are divisible by 11, but 0 is divisible by 11.) I**n base N, if alternate digits are added and subtracted, the divisibility of the result is the same as the divisibility of factors of N + 1.**

The example will come from base 11. We look at factors of 11 + 1 = 12.

22 is divisible by 2, but not by 4, 6, or 12. 22 base 10 is 20 base 11. 2 — 0 = 2, which is divisible by 2, but not 4 or 6.

102 base ten is divisible by 6, but not 4. In base 11, it is 93. 9 — 3 = 6, which is divisible by 6, but not 4.

2628 base ten is divisible by 12. 2628 base 10 is 1A7A base 11. 1 — A + 7 — A = 1 — 10 + 7 — 10 = —12.

Thus it is divisible by 12.

There is a nice website which converts between various bases.

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