2015 AMC 12A Problem 10

Problem 10 of 25EasierAlgebra Number Theory

Integers

x

and

y

with

x \gt y \gt 0

satisfy

x+y+xy = 80.

What is

x?

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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.

Concepts: Simon’s Favorite Factoring Trick · Diophantine Equation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.