Theorem: Let b be an integer. Then an integer a is divisible by (b-1) if and only if the sum of digits in its expansion (a)b is divisible by (b-1)
The proof is based on Modulo Arithmetic. We write a=b mod N iff (a-b) is divisible by N or, in other words, when a and b have the same remainder when divided by N. Such a and b are said to be congruent modulo N. From the definition one can easily derive several properties. For example
If a=b mod N and c=d mod N then (a+c)=(b+d) mod N
If a=b mod N and c=d mod N then ac=bd mod N
The criterion of divisibility by 9 follows from the fact that for every n 10n=1 mod 9. This, in turn, is a straightforward consequence of 10=1 mod 9.