2021 Fall AMC 10B Problems/Problem 22

Problem

For each integer $n\geq 2$ , let $S_n$ be the sum of all products $jk$ , where $j$ and $k$ are integers and $1\leq j<k\leq n$ . What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$ ? $\textbf{(A)}\ 196\qquad\textbf{(B)}\ 197\qquad\textbf{(C)}\ 198\qquad\textbf{(D)}\ 199\qquad\textbf{(E)}\ 200$

Solution 1

To get from $S_n$ to $S_{n+1}$ , we add $1(n+1)+2(n+1)+\cdots +n(n+1)=(1+2+\cdots +n)(n+1)=\frac{n(n+1)^2}{2}$ .

Now, we can look at the different values of $n$ mod $3$ . For $n\equiv 0\pmod{3}$ and $n\equiv 2\pmod{3}$ , then we have $\frac{n(n+1)^2}{2}\equiv 0\pmod{3}$ . However, for $n\equiv 1\pmod{3}$ , we have $\frac{1\cdot {2}^2}{2}\equiv 2\pmod{3}.$

Clearly, $S_2\equiv 2\pmod{3}.$ Using the above result, we have $S_5\equiv 1\pmod{3}$ , and $S_8$ , $S_9$ , and $S_{10}$ are all divisible by $3$ . After $3\cdot 3=9$ , we have $S_{17}$ , $S_{18}$ , and $S_{19}$ all divisible by $3$ , as well as $S_{26}, S_{27}, S_{28}$ , and $S_{35}$ . Thus, our answer is $8+9+10+17+18+19+26+27+28+35=27+54+81+35=162+35=\boxed{\mathrm{(B)}\ 197}$ . -BorealBear

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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2021 Fall AMC 10B Problems/Problem 22

Problem

Solution 1

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