Answer
0, 1, 5, 6
Work Step by Step
Consider $n$ an integer. We can write it in the form:
$n=10a+b$, where $b$ is its last digit.
Compute $n^4$:
$n^4=(10a+b)^4=10,000a^4+4000a^3b+600a^2b^2+40ab^3+b^4$.
The first 4 terms of this sum are numbers which end with a zero, therefore the last digit of $n^4$ is the last digit of $b^4$.
Compute $b^4$ for the numbers 0,1,2,...,8,9:
$0^4=0$
$1^4=1$
$2^4=16$
$3^4=81$
$4^4=256$
$5^4=625$
$6^4=1296$
$7^4=2401$
$8^4=4096$
$9^4=6561$
The last digit of $b^4$ is in the set $\{0,1,5,6\}$, so the fourth power of an integer ends in 0, 1, 5 or 6.
