Difference between revisions of "Divisibility rules/Rule for 2 and powers of 2 proof"
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* [[Divisibility rules | Back to divisibility rules]] | * [[Divisibility rules | Back to divisibility rules]] | ||
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Revision as of 20:26, 6 March 2014
A number is divisible by if the last digits of the number are divisible by .
Proof
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 .