2015 AMC 12A Problem 10 — Solution | LIVE by Po-Shen Loh

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2015 AMC 12A Problem 10

Below is the professionally curated solution for Problem 10 of the 2015 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:Simon’s Favorite Factoring Trick Diophantine Equation

Difficulty rating: 1530

10.

Integers $x$ and $y$ with $x \gt y \gt 0$ satisfy $x+y+xy = 80.$ What is $x?$

$8$

$10$

$15$

$18$

$26$

Solution:

Adding $1$ to both sides and factoring gives $(x+1)(y+1) = 81 = 3^4.$

Because $x$ and $y$ are distinct positive integers with $x \gt y,$ the only possibility is $x+1 = 3^3 = 27$ and $y+1 = 3.$ Therefore $x = 26.$

Thus, the correct answer is E.

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