Congruence and Remainders

For all integers and , if and only if and have the same remainder when divided by

For all integers and , with , if and only if has remainder when divided by

Determine the remainder when is divided by

Observe that .
From the propositions CAM and CP, we obtain

Since , we conclude from the proposition Congruent to Remainder that the remainder when is divided by is equal to

What is the last digit of ?

The last decimal digit of any non-negative integer is precisely equal to the remainder when is divided by . So we will work with modulo 10.

Observe that .
Using CAM and CP, we get

Since , we conclude from CTR that the remainder when is divided by is , and hence, the last decimal digit is

For all non-negative integers , is divisible by if and only if the sum of the digits in the decimal representation of is divisible by

For all non-negative integers , is divisible by if and only if is divisible by , where

• is the sum of the digits of even powers of 10 in the decimal representation of
• is the sum of the digits of odd powers of 10 in the decimal representation of