Problem 23 · 2017 AMC 8
Hard
Number Theory
factorizationarithmetic-sequence
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
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Answer: C — 25 miles.
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Hint 1 of 2
She rides 60 minutes each day. For the miles to come out whole, the minutes-per-mile must divide evenly into 60 — so each day's pace is a divisor of 60. That single fact is the whole problem.
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Hint 2 of 2
Now it's a treasure hunt: among the divisors of 60, find four that increase by exactly 5 each step. Listing them shows there's only one such run.
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Approach: minutes-per-mile must divide 60
- Each day she rides 60 minutes, and miles = 60 ÷ (minutes per mile). For that to be a whole number, the minutes-per-mile must be a divisor of 60. Translating 'integer miles' into 'divides 60' is the key move.
- Divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The only four that form an arithmetic run with common difference 5 are 5, 10, 15, 20.
- Total miles: 60/5 + 60/10 + 60/15 + 60/20 = 12 + 6 + 4 + 3 = 25.
- Why this transfers: 'whole number of X per fixed total' is almost always a disguised divisibility condition — convert it to 'must divide the total' and the search space shrinks to a short divisor list.
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