AMC 10 · 2013 · #17
Grade 6 number-theoryDaphne 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?
Pick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Three friends visit on fixed cycles: Alice every 3rd day, Beatrix every 4th day, and Claire every 5th day. All three visited on day 0 (yesterday). Over the next 365 days, count the days on which exactly two of the three friends visit.
Givens: Alice visits every 3 days, Beatrix every 4 days, Claire every 5 days.; All three visited together on the day before the period starts (day 0).; The period covers the next 365 days, days 1 through 365.
Unknowns: The number of days in the 365-day period on which exactly two friends visit.
Understand
Restated: Three friends visit on fixed cycles: Alice every 3rd day, Beatrix every 4th day, and Claire every 5th day. All three visited on day 0 (yesterday). Over the next 365 days, count the days on which exactly two of the three friends visit.
Givens: Alice visits every 3 days, Beatrix every 4 days, Claire every 5 days.; All three visited together on the day before the period starts (day 0).; The period covers the next 365 days, days 1 through 365.
Plan
Primary tool: #7 Identify Subproblems
Secondary: #16 Change Focus / Count the Complement
"Exactly two" breaks into three separate pair problems: Alice-Beatrix, Alice-Claire, and Beatrix-Claire. Two friends share a day only when it is a common multiple of their cycles, so each pair meets on multiples of a least common multiple. Counting each pair, then removing the days when all three actually show up, gives the exactly-two total.
Execute — Answer: B
6.NS.B.4 Step 1 Turn each pair into an LCM
- Two friends visit the same day only when that day is a multiple of both of their cycles, which is the least common multiple of the two intervals.
- Alice and Beatrix share a day every LCM(3,4)=12 days, Alice and Claire every LCM(3,5)=15 days, and Beatrix and Claire every LCM(4,5)=20 days.
💡 A shared visit day has to fit both friends' cycles at once, so it lands on their LCM.
6.NS.B.2 Step 2 Count each pair's meetings
- In 365 days, count how many multiples of each LCM occur by dividing 365 by the LCM and dropping the remainder.
- Alice and Beatrix meet on multiples of 12, Alice and Claire on multiples of 15, and Beatrix and Claire on multiples of 20.
💡 The number of multiples of k up to 365 is just how many whole times k fits into 365.
6.NS.B.4 Step 3 Find the all-three days
- All three friends land on the same day only when it is a multiple of all three cycles, which is LCM(3,4,5)=60.
- Count these across the 365 days.
💡 A day that suits all three cycles must be a multiple of their combined LCM, 60.
4.OA.A.3 Step 4 Remove the triple days from each pair
- Every multiple of 60 is also a multiple of 12, 15, and 20, so those 6 days got counted inside all three pair totals even though all three friends visit then, not exactly two.
- Subtract the 6 triple days from each pair count, then add the three adjusted counts together.
- The total is 54, which is choice (B).
💡 A day only counts as exactly two if the third friend stays away, so the all-three days must be stripped out of every pair.
6.NS.B.4 Two friends visit the same day only when that day is a multiple of both of their 6.NS.B.2 In 365 days, count how many multiples of each LCM occur by dividing 365 by the L 6.NS.B.4 All three friends land on the same day only when it is a multiple of all three c 4.OA.A.3 Every multiple of 60 is also a multiple of 12, 15, and 20, so those 6 days got c Review
Reasonableness: Check with one clean 60-day block, since the pattern repeats every 60 days and 365 covers six full blocks (360 days) with 5 extra. In one block, pairs meet 5, 4, and 3 times, but the day-60 triple meeting sits in all three, so exactly-two days are (5-1)+(4-1)+(3-1)=4+3+2=9. Over six blocks that is 9x6=54, and the leftover days 361-365 add none, matching 54.
Alternative: Count exactly-two days block by block instead of pair by pair: each 60-day block contributes 9 exactly-two days, and 365 days hold six complete blocks, giving 6x9=54 (B) with no overlap left to adjust.
CCSS standards used (min grade 6)
6.NS.B.4Find greatest common factor and least common multiple of two numbers (Finding the meeting cycle of each pair (12, 15, 20) and of all three (60) as least common multiples.)6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm (Counting multiples of 12, 15, 20, and 60 up to 365 by dividing.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Subtracting the 6 all-three days from each pair count and adding the results to reach 54.)
⭐ Two friends meet on the LCM of their cycles, so count each pair's meetings and take out the days when all three show up.
⭐ Two friends meet on the LCM of their cycles, so count each pair's meetings and take out the days when all three show up.
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