How are divisibility tests derived?

[soandos]

what is the proof for divisibility tests in general, and in particular how are they derived.
for example, i know that in base m, if the sum of the digits are a multiple of m-1, then m-1 is a factor. how tests like alternating digits having equal sums found?
does a test exist for every number?

 

[CRGreathouse]

what is the proof for divisibility tests in general, and in particular how are they derived.
for example, i know that in base m, if the sum of the digits are a multiple of m-1, then m-1 is a factor. how tests like alternating digits having equal sums found?
does a test exist for every number?

Divisibility tests in a given base are just special cases of modular arithmetic. Examples (base 10) for adding digits and alternating digits: 10a + b = a + b (mod 9), 100a + 10b + c = a - b + c (mod 11).