How many odd days are there in the 300 years from 1701 to 2000 (both years included)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2
An ordinary year has 1 odd day and a leap year 2. A century year is a leap year only if it is divisible by 400, so 1800 and 1900 are ordinary years but 2000 is a leap year. Work century by century.
1701 to 1800: 24 leap years and 76 ordinary years, \(76+2\times24=124=17\times7+5\), 5 odd days
1801 to 1900: also 24 leap years and 76 ordinary years, 5 odd days
1901 to 2000: 2000 is a leap year, so 25 leap years and 75 ordinary years, \(75+2\times25=125=17\times7+6\), 6 odd days
Total \(=5+5+6=16=2\times7+2\), so 2 odd days
Why the other options are incorrect
Option (2) 1 — Incorrect. This uses the rule “300 years give 1 odd day”, which holds only when none of the three century years is a leap year; here 2000 is one.
Option (3) 0 — Incorrect. 0 odd days belongs to a full 400-year cycle, not to 300 years.
Option (4) 3 — Incorrect. This treats 1800 as a leap year too; 1800 is not divisible by 400.