AMC8 2019

AMC8 2019 · Q23

AMC8 2019 · Q23. It mainly tests Fractions, Diophantine equations (integer solutions).

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members?

在 Euclid 高中最后一场篮球比赛后，确定球队得分中 Alexa 得了 $\frac{1}{4}$，Brittany 得了 $\frac{2}{7}$。Chelsea 得了 15 分。其他 7 名队员每人得分不超过 2 分。其他 7 名队员的总得分是多少？

(A) 10 10

(B) 11 11

(D) 13 13

(E) 14 14

Answer

Correct choice: (B)

正确答案：（B）

Solution

Given the information above, we start with the equation $\frac{t}{4}+\frac{2t}{7} + 15 + x = t$, where $t$ is the total number of points scored and $x\le 14$ is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation $x+15 = \frac{13}{28}t$, or $28x+28\cdot 15=13t$. Since $t$ is necessarily divisible by 28, let $t=28u$ where $u \ge 0$ and divide by 28 to obtain $x + 15 = 13u$. Then, it is easy to see $u=2$ ($t=56$) is the only candidate remaining, giving $x=\boxed{\textbf{(B)} 11}$.

根据以上信息，我们从方程 $\frac{t}{4}+\frac{2t}{7} + 15 + x = t$ 开始，其中 $t$ 是总得分，$x\le 14$ 是其余 7 名队员的得分，我们可以简化为不定方程 $x+15 = \frac{13}{28}t$，或 $28x+28\cdot 15=13t$。由于 $t$ 必须能被 28 整除，令 $t=28u$ 其中 $u \ge 0$，除以 28 得到 $x + 15 = 13u$。然后，很容易看出 $u=2$ ($t=56$) 是唯一剩余候选，得到 $x=\boxed{\textbf{(B)} 11}$。

Topics