Calendar: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

The odd-days method assigns each remainder from 0 to 6 to one day of the week. The standard convention starts the count at Sunday. So a remainder of 0 corresponds to Sunday, 1 to Monday, 2 to Tuesday, and so on. The count continues up to a remainder of 6, which corresponds to Saturday. Check: this gives exactly 7 remainders (0 through 6) for the 7 days of the week.

A century year is any year that is a multiple of 100. A year that is NOT a multiple of 100 is called a non-century year. For a non-century year, the /400 test never applies. Such a year is a leap year exactly when it is divisible by 4. Example check: 2024 is not a multiple of 100, and 2024 ÷ 4 = 506 exactly, so 2024 is a leap year.

A non-century year is a leap year if it is divisible by 4. A century year (divisible by 100) is a leap year only if it is also divisible by 400. All four options here — 2000, 1900, 2100, 1800 — are century years, so the /400 test decides each one. 2000 ÷ 400 = 5 exactly, so 2000 is divisible by 400 and is a leap year. 1800 ÷ 400 = 4.5, 1900 ÷ 400 = 4.75, and 2100 ÷ 400 = 5.25 — none are whole numbers. So 1800, 1900, and 2100 are NOT divisible by 400, and none of them is a leap year.

A week has 7 days, so a full week never changes the day of the week. A leap year has 366 days, one more than an ordinary year because of 29 February. 366 ÷ 7 = 52 remainder 2, so 366 days = 52 complete weeks (364 days) plus 2 extra days. So a leap year has exactly 2 odd days. Check: 52×7 + 2 = 366, which matches.

A week has 7 days, so a full week never changes the day of the week. An ordinary year has 365 days. 365 ÷ 7 = 52 remainder 1, so 365 days = 52 complete weeks (364 days) plus 1 extra day. That leftover 1 day is called an odd day. So an ordinary year has exactly 1 odd day. Check: 52×7 + 1 = 365, which matches.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 1955 falls on a Saturday, and 1955 is an ordinary year. Between 1955 and 1966 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 1966 falls on the same day, Saturday, satisfying condition one. 1966 is an ordinary year, matching 1955’s type, satisfying condition two. 1966 is the first later year meeting both conditions, so the calendar for 1955 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 1971 falls on a Friday, and 1971 is an ordinary year. Between 1971 and 1982 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 1982 falls on the same day, Friday, satisfying condition one. 1982 is an ordinary year, matching 1971’s type, satisfying condition two. 1982 is the first later year meeting both conditions, so the calendar for 1971 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 2040 falls on a Sunday, and 2040 is a leap year. Between 2040 and 2068 there are 21 ordinary year(s) and 7 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (21×1 + 7×2) mod 7 = 0 day(s). A shift of 0 means 1 January 2068 falls on the same day, Sunday, satisfying condition one. 2068 is a leap year, matching 2040’s type, satisfying condition two. 2068 is the first later year meeting both conditions, so the calendar for 2040 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 1972 falls on a Saturday, and 1972 is a leap year. Between 1972 and 2000 there are 21 ordinary year(s) and 7 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (21×1 + 7×2) mod 7 = 0 day(s). A shift of 0 means 1 January 2000 falls on the same day, Saturday, satisfying condition one. 2000 is a leap year, matching 1972’s type, satisfying condition two. 2000 is the first later year meeting both conditions, so the calendar for 1972 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 2054 falls on a Thursday, and 2054 is an ordinary year. Between 2054 and 2065 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 2065 falls on the same day, Thursday, satisfying condition one. 2065 is an ordinary year, matching 2054’s type, satisfying condition two. 2065 is the first later year meeting both conditions, so the calendar for 2054 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 1967 falls on a Sunday, and 1967 is an ordinary year. Between 1967 and 1978 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 1978 falls on the same day, Sunday, satisfying condition one. 1978 is an ordinary year, matching 1967’s type, satisfying condition two. 1978 is the first later year meeting both conditions, so the calendar for 1967 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 2015 falls on a Thursday, and 2015 is an ordinary year. Between 2015 and 2026 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 2026 falls on the same day, Thursday, satisfying condition one. 2026 is an ordinary year, matching 2015’s type, satisfying condition two. 2026 is the first later year meeting both conditions, so the calendar for 2015 repeats there.

Two years share the same calendar only if two conditions both hold. First, 1 January falls on the same day of the week in both years. Second, both years are the same type — both leap or both ordinary — so February lines up too. 1 January 1975 falls on a Wednesday, and 1975 is an ordinary year. Between 1975 and 1986 there are 8 ordinary year(s) and 3 leap year(s). An ordinary year contributes 1 odd day and a leap year contributes 2. The total shift is (8×1 + 3×2) mod 7 = 0 day(s). A shift of 0 means 1 January 1986 falls on the same day, Wednesday, satisfying condition one. 1986 is an ordinary year, matching 1975’s type, satisfying condition two. 1986 is the first later year meeting both conditions, so the calendar for 1975 repeats there.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1757 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1757 ÷ 4 = 439.25, not a whole number, so 1757 is not divisible by 4. So 1757 is an ordinary year, not a leap year. Check: 1757 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 2054 is not divisible by 100, so it is not a century year, and only the /4 test applies. 2054 ÷ 4 = 513.5, not a whole number, so 2054 is not divisible by 4. So 2054 is an ordinary year, not a leap year. Check: 2054 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 2019 is not divisible by 100, so it is not a century year, and only the /4 test applies. 2019 ÷ 4 = 504.75, not a whole number, so 2019 is not divisible by 4. So 2019 is an ordinary year, not a leap year. Check: 2019 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1731 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1731 ÷ 4 = 432.75, not a whole number, so 1731 is not divisible by 4. So 1731 is an ordinary year, not a leap year. Check: 1731 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1928 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1928 ÷ 4 = 482 exactly, so 1928 is divisible by 4. So 1928 is a leap year. Check: 1928 has 366 days, including 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1858 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1858 ÷ 4 = 464.5, not a whole number, so 1858 is not divisible by 4. So 1858 is an ordinary year, not a leap year. Check: 1858 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1825 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1825 ÷ 4 = 456.25, not a whole number, so 1825 is not divisible by 4. So 1825 is an ordinary year, not a leap year. Check: 1825 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1942 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1942 ÷ 4 = 485.5, not a whole number, so 1942 is not divisible by 4. So 1942 is an ordinary year, not a leap year. Check: 1942 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 1826 is not divisible by 100, so it is not a century year, and only the /4 test applies. 1826 ÷ 4 = 456.5, not a whole number, so 1826 is not divisible by 4. So 1826 is an ordinary year, not a leap year. Check: 1826 has 365 days, no 29 February.

The leap year rule: a year divisible by 4 is a leap year. The exception is a century year (divisible by 100) that fails the /400 test. 2095 is not divisible by 100, so it is not a century year, and only the /4 test applies. 2095 ÷ 4 = 523.75, not a whole number, so 2095 is not divisible by 4. So 2095 is an ordinary year, not a leap year. Check: 2095 has 365 days, no 29 February.

The 1st of January 1972 falls on a Saturday. Wednesday is 4 day(s) after Saturday in the week, so the first Wednesday of the month is the 5th. A week has 7 days, so every following Wednesday falls exactly 7 days later. Adding 7 repeatedly within the 31-day month gives 5th, 12th, 19th, 26th. Check: consecutive dates in 5th, 12th, 19th, 26th are each 7 days apart, all within the month.

The 1st of October 1956 falls on a Monday. Saturday is 5 day(s) after Monday in the week, so the first Saturday of the month is the 6th. A week has 7 days, so every following Saturday falls exactly 7 days later. Adding 7 repeatedly within the 31-day month gives 6th, 13th, 20th, 27th. Check: consecutive dates in 6th, 13th, 20th, 27th are each 7 days apart, all within the month.