Problem 17 · AMC 8 Stretch
Core
Number Theory
Logic & Word Problems
logical-reasoningpattern-recognition
In a leap year, January has \(31\) days, February \(29\), March \(31\). Using the fact that a week repeats every \(7\) days, January 1 and April 1 fall on the same weekday because the number of days from January 1 to April 1 is a multiple of \(7\). How many days is that?
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Answer: 91 days (= 7 x 13)
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Hint 1 of 4
Two dates land on the same weekday when the number of days between them is a multiple of \(7\). So count the days from January 1 to April 1.
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Hint 2 of 4
The days from January 1 to April 1 equal the lengths of January, February, and March added together.
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Hint 3 of 4
In a leap year, January has \(31\) days, February has \(29\), March has \(31\). Add them up.
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Approach: Count days, then check divisibility by 7
- Two dates fall on the same weekday exactly when the number of days between them divides evenly by \(7\), since the weekday pattern repeats every \(7\) days.
- From January 1 to April 1 the days you pass through are all of January, February, and March: \(31 + 29 + 31 = 91\) days.
- Divide by \(7\): \(91 = 7 \times 13\), an exact multiple with no remainder. So April 1 lands on the same weekday as January 1.
- Bonus: April 1 to July 1 is \(30 + 31 + 30 = 91\) days too, so July 1 also matches — that is why January, April, and July all start on the same weekday in a leap year.
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