How to Calculate Odd Days for Calendar Problems?
Odd Days means the remainder left when the total number of days is divided by 7. This remainder shows how many weekdays the calendar moves forward. Learn the odd days formula, leap-year rules, month-wise calculations and direct methods for solving calendar-based aptitude questions.
What Are Odd Days?
The formula is Odd Days = Total Days mod 7. A remainder of 0 causes no weekday shift, while remainders 1 to 6 shift the weekday forward by 1 to 6 days. For example, 17 days contain 2 complete weeks and 3 odd days, so the weekday moves forward by 3 days.
Odd Days Video Lessons
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Calendar Odd Days and Day Calculation
Learn how to calculate odd days and determine the day of the week for given dates using calendar-based aptitude methods.
Odd Days Formula & Tricks
Important Formulas
Only the remainder after division by 7 is counted as odd days.
Each ordinary year has 365 days, and 365 leaves 1 odd day after division by 7.
Here, n is the number of years and L is the number of leap years in the period. Each leap year contributes 2 odd days instead of 1.
This Gregorian-calendar formula counts years divisible by 4, excluding century years not divisible by 400.
Move forward by the remainder when the number of intervening days is divided by 7.
Quick Tricks
A 31-day month gives 3 odd days, a 30-day month gives 2, February in an ordinary year gives 0, and February in a leap year gives 1.
In the Gregorian calendar, 100 years have 5 odd days, 200 years have 3, 300 years have 1 and 400 years have 0 odd days.
To find the weekday of the dth day of a month, add the days in all previous months and d - 1, not d.
Odd Days Concepts
Odd Days in Ordinary and Leap Years
An ordinary year has 365 days, and 365 = 52 × 7 + 1. A leap year has 366 days, and 366 = 52 × 7 + 2. Therefore, the weekday of 1 January shifts by 1 day after an ordinary year and by 2 days after a leap year.
Leap-Year Rule
Thus, 2024 is a leap year, 1900 is not a leap year, and 2000 is a leap year. February has 29 days in a leap year and 28 days otherwise.
Odd Days in Months
A 31-day month contributes 3 odd days, a 30-day month contributes 2, February contributes 0 in an ordinary year and 1 in a leap year. The month lengths are useful when finding the weekday of a date.
Finding the Weekday of a Date
For a date within the same year, add the days in all completed months and the date minus 1. For dates in different years, first calculate the odd days in the complete years and then add the month and date contributions.
Odd Days Data Table
Use the structured data below while following the explanation and examples.
Month-Wise Odd Days
Odd days contributed by each month according to the number of days it contains.
| Month type | Number of days | Odd days |
|---|---|---|
| 31-day month | 31 | 3 |
| 30-day month | 30 | 2 |
| February in an ordinary year | 28 | 0 |
| February in a leap year | 29 | 1 |
Note: The odd days are the remainders after division by 7.
Odd Days in Century Blocks
Odd days in Gregorian-calendar blocks of complete centuries.
| Period | Total odd days |
|---|---|
| 100 years | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
Note: A 400-year Gregorian cycle contains exactly 0 odd days.
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Odd Days Revision Points
Use these rules to revise calendar calculations quickly.
- Odd Days = total days mod 7.
- 365 days give 1 odd day; 366 days give 2 odd days.
- A leap year is divisible by 4, except century years not divisible by 400.
- A 31-day month gives 3 odd days; a 30-day month gives 2.
- February gives 0 odd days in an ordinary year and 1 in a leap year.
- In a Gregorian calendar, 100, 200, 300 and 400 years give 5, 3, 1 and 0 odd days respectively.
- For the dth date, add d - 1 after counting all completed months.
Odd Days FAQs
What is the odd days formula?
Odd Days = Total number of days mod 7. It is the remainder obtained after dividing the total days by 7.
How many odd days are there in 15 days?
15 = 2 × 7 + 1, so there is 1 odd day.
How many odd days does a leap year have?
A leap year has 366 days, and 366 = 52 × 7 + 2. Therefore, it has 2 odd days.
Why does 1900 not count as a leap year?
Although 1900 is divisible by 4, it is a century year and is not divisible by 400. Therefore, 1900 is an ordinary year.
How many odd days are there in 100 years?
In 100 years there are 100 ordinary-year contributions plus 24 leap-year corrections, giving 36524 days. Since 36524 mod 7 = 5, there are 5 odd days.
How are odd days used to find the weekday of 15 August 2023 if 1 January 2023 was Sunday?
Days before August are 212 and the offset for the 15th day is 14. Thus, 212 + 14 = 226, and 226 mod 7 = 2. Two days after Sunday is Tuesday.