AMC12 2015 A
AMC12 2015 A · Q10
AMC12 2015 A · Q10. It mainly tests Manipulating equations, Primes & prime factorization.
Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 80$. What is $x$?
整数 $x$ 和 $y$ 满足 $x > y > 0$ 且 $x + y + xy = 80$。$x$ 是多少？
(A)
8
8
(B)
10
10
(C)
15
15
(D)
18
18
(E)
26
26
Answer
Correct choice: (E)
正确答案：（E）
Solution
Answer (E): Adding 1 to both sides of the equation and factoring yields \((x+1)(y+1)=81=3^4\). Because \(x\) and \(y\) are distinct positive integers and \(x>y\), the only possibility is that \(x+1=3^3=27\) and \(y+1=3^1=3\). Therefore \(x=26\).
答案（E）：在等式两边都加 1 并因式分解，得到 \((x+1)(y+1)=81=3^4\)。因为 \(x\) 和 \(y\) 是不同的正整数且 \(x>y\)，唯一可能是 \(x+1=3^3=27\)，\(y+1=3^1=3\)。因此 \(x=26\)。
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