Julia Thompson wrote:
>
> In base N, to check to see if a number is divisible by N-1, just add the
> digits, and if their sum is divisible by N-1, the number itself is.  So
> in base 10, if the sum of the digits of a number add up to 9 or 18 or
> 27, etc., the number is divisible by 9.
>
> If N-1 is a square, a similar divisibility test will work on sqrt(N-1).
> So if the sum of digits of a number in base 10 is divisible by 3, the
> number itself is divisible by 3.
>
No, the correct form of the test is: if N-1 is not a prime number,
then the same test works for the _divisors_ of N-1.

And there's also a simple test for N+1 and (N+1)'s divisors:
if the sum of the even-ordered digits and the sum of the odd-ordered
digits differ by a multiple of the divisor under test, then the
number is divisible by it. For example, 8074 is divisible by 11
in base 10.

> Base 12 has easier divisibility tests for more numbers, though.
>
Yes, but we gain useless numbers, like 13, and lose useful numbers,
like 5.

Alberto Monteiro

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