Skip to main content

2013 AMC 10A Problem 17

Problem 17 of 25IntermediateNumber TheoryCombinatorics

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 365365-day period will exactly two friends visit her?

Answer choices

Show solution

Solution

Pairwise visits occur every lcm⁡(3,4)=12\operatorname{lcm}(3,4)=12, lcm⁡(3,5)=15\operatorname{lcm}(3,5)=15, and lcm⁡(4,5)=20\operatorname{lcm}(4,5)=20 days. In the next 365365 days, these pair counts are ⌊36512⌋=30\left\lfloor\frac{365}{12}\right\rfloor=30, ⌊36515⌋=24\left\lfloor\frac{365}{15}\right\rfloor=24, and ⌊36520⌋=18\left\lfloor\frac{365}{20}\right\rfloor=18. All three visit every 6060 days, which happens ⌊36560⌋=6\left\lfloor\frac{365}{60}\right\rfloor=6 times. Subtracting those days from each pair count gives (30−6)+(24−6)(30-6)+(24-6) +(18−6)=54+(18-6)=54. Thus, B is the correct answer.
AoPS wiki

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

More practice