n-ary representation and divisibility


Recently I have an observation about algorithms on divisibility of
integers by prime numbers. People have developed many of these, e.g.

(1)For p=3: add up all digits and see if the sum divides by 3.

(2)For p=7:split the digits two by two from the leftmost digit; then 2
times the leftmost group and add on the second leftmost group, and so
on, till see if the sum divides by 7.

(3)For p=11: split the digits two by two from the leftmost digit;
them add the leftmost group on the second leftmost group, so on, till
see if the final sum divides by 11.

But we perform all these algorithms mostly under decimal representation
of the integer. What about for different n-ary representation?

I find that: For integer a and fixed prime p, if we write a into m-ary
representation, where m and 10 are congruent mod p, then the same
algorithm that works under decimal representation, could still applies
for m-ary representation of a. I've checked tens of numbers and the
first several primes, and the conclusion goes true, but i dont know how
to prove it or disprove it.

Could anyone help me?

.

Relevant Pages

  • Re: n-ary representation and divisibility
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