Discussion

What is the unit digit of the sum \(1^5 + 2^5 + 3^5 + ... + 20^5\)?

Answer: A

The unit digit of \(n^5\) is the same as the unit digit of n. This is because the cycle length for all digits' powers divides 4, and \(5 \equiv 1 \pmod 4\). So, the unit digit of \(k^5\) is the same as the unit digit of k. We need to find the unit digit of the sum \(1+2+3+...+20\). The sum is \(\frac{20(21)}{2} = 210\). The unit digit of this sum is 0.