What Day of the Week in Year N-1

Number Theory Level 3

In year N, the 300th day of the year is a Tuesday. In year N+1, the 200th day is also a Tuesday. On what day of the week did the 100th day of the year N-1 occur?

Sunday Wednesday Thursday Saturday Friday Tuesday Monday

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2 solutions

Chew-Seong Cheong
Feb 11, 2020

If year N is not a leap year, then 200th day of year N+1 is 65 + 200 = 265 65+200 = 265 days away. To find the weekday, we calculate the remainder when the number of days is divided by 7, which is 265 ≡ 6 ≡ − 1 (mod 7) 265 \equiv 6 \equiv -1 \text{ (mod 7)} or a Monday. Therefore year N is a leap year with 366 days and year N-1 was not a leap year. Then the 100th day of year N-1 is − 300 − 265 = − 565 -300-265 = -565 days away which is − 565 ≡ − 5 ≡ 2 (mod 7) -565 \equiv - 5 \equiv 2 \text{ (mod 7)} a Thursday .

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One such recent example was Oct 26,2004 (300th day on Tues), Jul 19, 2005 (200th day on Tues), & Apr 10, 2003 (100th day on Thurs).

tom engelsman - 1 year, 4 months ago

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