Divisibility rules/Rule for 5 and powers of 5 proof
A number is divisible by if the last digits are divisible by that power of 5.
An understanding of basic modular arithmetic is necessary for this proof.
Let the base-ten representation of be where the are digits for each and the underline is simply to note that this is a base-10 expression rather than a product. If has no more than digits, then the last digits of make up itself, so the test is trivially true. If has more than digits, we note that:
Taking this we have
because for , . Thus, is divisible by if and only if
is. But this says exactly what we claimed: the last digits of are divisible by if and only if is divisible by .