Question

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?

$48$

$54$

$60$

$66$

$72$

Solution:

Pairwise visits occur every $\operatorname{lcm}(3,4)=12$ , $\operatorname{lcm}(3,5)=15$ , and $\operatorname{lcm}(4,5)=20$ days.

In the next $365$ days, these pair counts are $\left\lfloor\frac{365}{12}\right\rfloor=30$ , $\left\lfloor\frac{365}{15}\right\rfloor=24$ , and $\left\lfloor\frac{365}{20}\right\rfloor=18$ .

All three visit every $60$ days, which happens $\left\lfloor\frac{365}{60}\right\rfloor=6$ times. Subtracting those days from each pair count gives $(30-6)+(24-6)+(18-6)=54$ .

Thus, B is the correct answer.

2013 AMC 10A Problem 17

Solution:

Problem 17 in Other Years