AMC10 2013 A

AMC10 2013 A · Q17

AMC10 2013 A · Q17. It mainly tests Basic counting (rules of product/sum), GCD & LCM.

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?

Daphne 定期被她的三位最好的朋友访问：Alice、Beatrix 和 Claire。Alice 每三天访问一次，Beatrix 每四天访问一次，Claire 每五天访问一次。三位朋友昨天都访问了 Daphne。在接下来的 365 天周期中，有多少天恰好有两位朋友访问她？

(A) 48 48

(B) 54 54

(D) 66 66

(E) 72 72

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Alice and Beatrix will visit Daphne together every $3\cdot 4=12$ days, so this will happen $\left\lfloor \frac{365}{12}\right\rfloor=30$ times. Likewise Alice and Claire will visit together $\left\lfloor \frac{365}{3\cdot 5}\right\rfloor=24$ times, and Beatrix and Claire will visit together $\left\lfloor \frac{365}{4\cdot 5}\right\rfloor=18$ times. However, each of these counts includes the $\left\lfloor \frac{365}{3\cdot 4\cdot 5}\right\rfloor=6$ times when all three friends visit. The number of days that exactly two friends visit is $(30-6)+(24-6)+(18-6)=54$.

答案（B）：Alice 和 Beatrix 每隔 $3\cdot 4=12$ 天会一起去拜访 Daphne，因此一年中会发生 $\left\lfloor \frac{365}{12}\right\rfloor=30$ 次。同样，Alice 和 Claire 一起去拜访的次数为 $\left\lfloor \frac{365}{3\cdot 5}\right\rfloor=24$ 次，Beatrix 和 Claire 一起去拜访的次数为 $\left\lfloor \frac{365}{4\cdot 5}\right\rfloor=18$ 次。不过，这些计数都包含了三人同时去拜访的 $\left\lfloor \frac{365}{3\cdot 4\cdot 5}\right\rfloor=6$ 次。恰好两位朋友去拜访的天数为 $(30-6)+(24-6)+(18-6)=54$。

Topics