Problem 10 · 1988 AJHSME
Medium
Number Theory
mod-7
Chris's birthday is on a Thursday this year. What day of the week will it be 60 days after her birthday?
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Answer: A — Monday.
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Hint 1 of 2
A whole week brings you right back to the same weekday — Thursday to Thursday. So most of those 60 days do nothing to the answer. What's the only part that matters?
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Hint 2 of 2
Pull as many full weeks (groups of 7) out of 60 as you can; only the leftover days actually move the weekday forward.
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Approach: throw away whole weeks, count only the leftover days
- Every 7 days lands on the same weekday, so the full weeks inside 60 days change nothing. 60 = 56 + 4 = eight full weeks plus 4 leftover days, and only those 4 matter.
- Step 4 days past Thursday: Friday, Saturday, Sunday, Monday.
- Why this transfers: any 'what comes after a big jump in a repeating cycle' question works this way — strip out whole cycles (weeks, here) and walk only the remainder. The same trick finds the day of the year, the hour on a clock, or a repeating decimal digit.
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