What is the unit digit of the sum 1^5 + 2^5 + 3^5 + … + 20^5 in number system aptitude?

Introduction / Context:
This question tests understanding of unit digit patterns in powers and how to sum them efficiently without computing the entire large numbers. Working with powers from 1^5 to 20^5 is too large to do directly, so pattern recognition in the last digit is the key idea.

Given Data / Assumptions:

• We consider the sum S = 1^5 + 2^5 + 3^5 + … + 20^5.
• The task is to find only the unit digit of S.
• All numbers involved are positive integers.

Concept / Approach:
For any integer n, the unit digit of n^5 depends only on the unit digit of n, and in fact for every digit d from 0 to 9, the unit digit of d^5 is the same as d. This means that 7^5 ends with 7, 3^5 ends with 3, and so on. Therefore, the unit digit of S equals the unit digit of the sum of the integers from 1 to 20. We can use the formula for the sum of the first n natural numbers and then look at the unit digit.

Step-by-Step Solution:
Observe the pattern for unit digits of fifth powers: 1^5 ends in 1, 2^5 ends in 2, 3^5 ends in 3, 4^5 ends in 4, 5^5 ends in 5, 6^5 ends in 6, 7^5 ends in 7, 8^5 ends in 8, 9^5 ends in 9, 10^5 ends in 0. The pattern repeats every 10 numbers, because the unit digit of n and n + 10 is the same. Therefore, the unit digit of n^5 equals the unit digit of n for all integers n. This means the unit digit of the whole sum S is the same as the unit digit of 1 + 2 + 3 + … + 20. Use the formula for the sum of the first n natural numbers: 1 + 2 + … + n = n(n + 1) / 2. For n = 20, we get sum = 20 × 21 / 2 = 10 × 21 = 210. The unit digit of 210 is 0. Therefore, the unit digit of 1^5 + 2^5 + 3^5 + … + 20^5 is also 0.

Verification / Alternative check:
We can group terms by unit digit. Numbers ending in 1, 2, …, 9, 0 appear exactly twice in the range 1 to 20: for example 1 and 11, 2 and 12, and so on, plus 10 and 20. Each pair contributes a combined unit digit equal to twice the digit. Summing all these contributions and then taking the unit digit again leads to the same result of 0.

Why Other Options Are Wrong:
5, 2, 4, 8: These would appear if one misunderstood the pattern for fifth powers or only added the first few terms. The correct approach uses the fact that n^5 has the same unit digit as n, which leads uniquely to 0.

Common Pitfalls:
A common error is to try to compute several powers directly instead of using the pattern, which is slow and prone to mistakes. Another mistake is to forget that the pattern repeats every 10 numbers and to assume a different cycle length. Remembering that fifth powers preserve the unit digit makes such questions very quick.

Final Answer:
The unit digit of the sum 1^5 + 2^5 + 3^5 + … + 20^5 is 0.