Question

整数 $x$ 和 $y$ 满足 $x \gt y \gt 0$ 且 $x+y+xy = 80$ 。 $x$ 是多少？

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

$8$

$10$

$15$

$18$

$26$

两边同时加 $1$ 并因式分解，得 $(x+1)(y+1) = 81 = 3^4.$

因为 $x$ 和 $y$ 是不同的正整数且 $x \gt y$ ，唯一可能是 $x+1 = 3^3 = 27$ 且 $y+1 = 3$ 。因此 $x = 26$ 。

因此，正确答案是 E。

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.