Divisibility rules for numbers in bases other than ten

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. In 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.) In 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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