Divisibility rules/Rule for 2 and powers of 2 proof
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 be the base-ten expression for
, where the
are digits.
Thus
![$N = 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10 a_1 + a_0.$](http://latex.artofproblemsolving.com/2/9/8/298c376ff3a0f4ffb08ae3357165f9f92c517586.png)
Taking mod
gives
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Thus, if the last digits of
are divisible by
then
is divisible by
.