2013 AMC 10A Problem 17

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

Answer: B

Concepts:least common multiple inclusion-exclusion counting integers in a range

Difficulty rating: 1420

Small hint:

Count pairwise meeting days using least common multiples

Big hint:

Subtract the triple-meeting days from each pair count

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.