Question

For each integer $n\ge2,$ let $S_n$ be the sum of all products $jk,$ where $j$ and $k$ are integers and $1\le j<k\le n.$ What is the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3?$

$\ 196$

$\ 197$

$\ 198$

$\ 199$

$\ 200$

Answer: B

Concepts:modular arithmetic summation pattern recognition

Difficulty rating: 1950

Solution:

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

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

Answer: B

Difficulty rating: 1950

Solution:

Problem 22 in Other Years