2005 iTest Problems/Problem 50

Problem

When $1^0 + 2^1 + 3^2 + ...+ 100^{99}$ is divided by $5$ , a remainder of $N$ is obtained such that $N$ is between $0$ and $4$ inclusive. Find $N$ .

Solution 1

We notice a pattern: since remainders when divided by 5 repeat in cycles of 4 and there are 5 possible remainders the remainders of terms repeat every 20 terms. The remainder we seek will be 5 times this sum. Since the sum we seek is a multiple of 5, the $N$ we seek is equal to $\boxed{0}$ .

2005 iTest (Problems, Answer Key)
Preceded by: Problem 49	Followed by: Problem 51
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2005 iTest Problems/Problem 50

Problem

Solution 1

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