# Difference between revisions of "2016 AMC 10B Problems/Problem 18"

## Problem

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

## Solution

Factorize $345=3\cdot 5\cdot 23$.

Suppose we take an odd number $k$ of consecutive integers, centred on $m$. Then $mk=345$ with $\tfrac12k. Looking at the factors of $345$, the possible values of $k$ are $3,5,15,23$ centred on $115,69,23,15$ respectively.

Suppose instead we take an even number $2k$ of consecutive integers, centred on $m$ and $m+1$. Then $k(2m+1)=345$ with $k\le m$. Looking again at the factors of $345$, the possible values of $k$ are $1,3,5$ centred on $(172,173),(57,58),(34,35)$ respectively.

Thus the answer is $\textbf{(E) }7$.

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