The number of odd days in 1600 years is 0.
Here's why:
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Understanding Odd Days: Odd days are the remaining days after dividing the total number of days by 7 (a week). These remaining days determine the day of the week for a specific date.
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Odd Days in a Century (100 Years): A normal year has 365 days (52 weeks + 1 day). A leap year has 366 days (52 weeks + 2 days). In a 100-year period, there are 76 normal years and 24 leap years (though this approximation is crucial - adjustments are needed because not every 4th year is a leap year - century years not divisible by 400 are NOT leap years). Thus, a century has (76 1) + (24 2) = 76 + 48 = 124 odd days. When we divide 124 by 7, we get 17 weeks and 5 odd days. Therefore, 100 years have 5 odd days.
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Odd Days in 400 Years: To understand 1600 years, knowing the odd days in 400 years is essential. In 400 years, the leap year exception comes into play. Every 400 years, the century year is a leap year. So we now have 100, 200, 300 and 400 years to consider.
- 100 years have 5 odd days.
- 200 years have (5 * 2) = 10 odd days, which is equivalent to 3 odd days (10 mod 7 = 3).
- 300 years have (5 * 3) = 15 odd days, which is equivalent to 1 odd day (15 mod 7 = 1).
- 400 years have (5 * 4) + 1 = 21 odd days (we add 1 because the 400th year is a leap year). 21 odd days is equivalent to 0 odd days (21 mod 7 = 0).
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Odd Days in 1600 Years: Since 400 years have 0 odd days, any multiple of 400 years will also have 0 odd days. 1600 is a multiple of 400 (1600 = 400 * 4). Therefore, 1600 years have 0 odd days.
