AMC12 2025 B

AMC12 2025 B · Q5

AMC12 2025 B · Q5. It mainly tests Systems of equations, Diophantine equations (integer solutions).

Positive integers $x$ and $y$ satisfy the equation $57x+ 22y = 400$. What is the least possible value of $x+y$?

正整数 $x$ 和 $y$ 满足方程 $57x+ 22y = 400$。$x+y$ 的最小可能值是多少？

(A) 10 10

(B) 11 11

(D) 14 14

(E) 15 15

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let $x+y = a$. Then we have the equation \[35x + 22a = 400.\] Because the other two terms are divisible by $5$, $22a$ must be divisible by $5$ too. Specifically, $a$ is divisible by $5$. Let $a=5b$ and substitute: \[35x + 110b = 400\] \[7x + 22b = 80.\] After some analysis, we find that if $b=3$, $x = 2$. In fact, this is the only solution with positive integer solutions. Therefore, $a = x+y = 5 \cdot 3 = \boxed{\textbf{(E) }15}$.

令 $x+y = a$。则方程为 \[35x + 22a = 400.\] 由于其余两项能被5整除，$22a$ 也须被5整除，即 $a$ 能被5整除。令 $a=5b$ 代入： \[35x + 110b = 400\] \[7x + 22b = 80.\] 分析后，当 $b=3$ 时 $x = 2$。这是唯一正整数解。因此 $a = x+y = 5 \cdot 3 = \boxed{\textbf{(E) }15}$。

Topics