Question

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 $3 ?$

$\ 196$

$\ 197$

$\ 198$

$\ 199$

$\ 200$

Solution:

When passing from $S_{n-1}$ to $S_n$ , the new terms are $jn$ for $1\le j\lt n$ . Their sum is $n(1+2+\cdots+(n-1))=\frac{n^2(n-1)}2.$

Modulo $3$ , this increment is $0$ when $n\equiv0$ or $1\pmod3$ , and is $2$ when $n\equiv2\pmod3$ .

Since $S_2=2$ , the sequence becomes divisible by $3$ after the third occurrence of a number congruent to $2\pmod3$ , namely at $n=8$ . Then $S_n$ stays divisible by $3$ for $n=8,9,10$ , and the same pattern repeats every $9$ in $n$ .

The ten least values are $8,9,10,17,18,19,26,27,28,35$ . Their sum is $197$ .

Thus, the answer is B .

2021 AMC 10B Fall Problem 22

Solution:

Problem 22 in Other Years