2009 UNCO Math Contest II Problems/Problem 2

Problem

(a) Let $Q_n=1^n+2^n$ . For how many $n$ between $1$ and $100$ inclusive is $Q_n$ a multiple of $5$ ?

(b) For how many $n$ between $1$ and $100$ inclusive is $R_n=1^n+2^n+3^n+4^n$ a multiple of 5?

Solution

(a) Looking at the units digits, we need the units digit of $Q_n$ to be either $0$ or $5$ . We know that $1^n$ will always have a units digit of $1$ . Looking at $2^n$ , however, cycles every four powers with units digits $2, 4, 8,$ and $6$ in that order. We see that we can only get a units digit of $5$ if we have $4$ as a units digit for $2^n$ , and there is no way to get $0$ as a units digit. Therefore, our answer is $\boxed{25}$ because the four units digits cycle $25$ times in the integers $1$ to $100$ .

(b) Similarly, $3^n$ cycles every four powers with units digits $3, 9, 7,$ and $1$ in that order. And $4^n$ cycles every two powers with units digits $4$ and $6$ . Together the units digit of their sum is $0$ for $n=1,2,3 \pmod 4$ , and $4$ for $N=0 \pmod 4$ . So the answer is $\boxed{75}$ .

2009 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2009 UNCO Math Contest II Problems/Problem 2

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