2016 AMC 12B Problem 16

Below is the professionally curated solution for Problem 16 of the 2016 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12B solutions, or check the answer key.

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Concepts:arithmetic sequence factor counting casework

Difficulty rating: 1800

16.

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

$1$

$3$

$5$

$6$

$7$

Solution:

A sum of consecutive integers equals the count times the median. For an odd number of terms, the median is an integer divisor of $345,$ giving runs of $3$ (median $115$ ), $5$ (median $69$ ), $15$ (median $23$ ), and $23$ (median $15$ ) terms. For an even number of terms $2k,$ the median is a half-integer, giving runs of $2,$ $6,$ and $10$ terms. Longer runs would force negative terms. This gives $4+3=7$ ways.

Thus, the correct answer is E.

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