Problem 24 · 2017 AMC 8
Hard
Number Theory
complementary-countingdivisibility
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
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Answer: D — 146 days.
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Hint 1 of 2
Don't track 365 days — the call pattern is periodic. After lcm(3,4,5) = 60 days all three are back in sync, so figure out one 60-day block and the rest is copy-paste.
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Hint 2 of 2
Count no-call days = 60 − (call days), and find call days by inclusion–exclusion: add the every-3, every-4, every-5 counts, subtract the overlaps (every-12, every-15, every-20), add back the triple (every-60).
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Approach: find one 60-day cycle by inclusion–exclusion, then tile
- The three grandchildren realign every lcm(3, 4, 5) = 60 days, so the whole year is just copies of one 60-day block. Solving one block solves everything.
- Call days in 60 by inclusion–exclusion: 20 (÷3) + 15 (÷4) + 12 (÷5) − 5 (÷12) − 4 (÷15) − 3 (÷20) + 1 (÷60) = 36. So no-call days = 60 − 36 = 24 per block.
- The year is 365 = 6×60 + 5 days. The six full blocks give 6 × 24 = 144 no-call days.
- Handle the 5 leftover days (days 361–365, which restart the cycle as days 1–5): day 1 no, day 2 no, day 3 call, day 4 call, day 5 call — 2 more no-call days.
- Total: 144 + 2 = 146.
- Why this transfers: periodic-event problems collapse to one lcm-length cycle; inclusion–exclusion handles the overlaps, and the only care needed is the partial cycle at the end.
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