Last Digit

Every number when put to the power of 5, has the same last digit of the number you put to the power of 5.

Therefore, you add all the numbers from 1 - 99 to get your answer.

Given we know 1 + 2 + 3 +.... + 100 = 5050, we subtract 100 to find 1 + 2 + 3 +.... + 99.

5050 - 100

= 495 0

$\phi(10)=10\times(1-\dfrac 12)\times(1-\dfrac 15)=4$
$1^5+2^5+\cdots+99^5\equiv 1+2+\cdots+99=4950\equiv 0(\text{mod }10)$