# Difference between revisions of "1993 AHSME Problems/Problem 16"

## Problem

Consider the non-decreasing sequence of positive integers $$1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots$$ in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 2\quad \text{(D) } 3\quad \text{(E) } 4$

## Solution

You want to find the largest integer $n$ that satisfies $\frac{n(n+1)}{2}<1993$.

By trial and error, the value of $n$ is $62$. Therefore, the next value of the sequence is $63$, and $63 \div 5$ has a remainder of $3$.

$\fbox{D}$