Odd days can be effectively counted concerning a particular year by using various mathematical techniques. Normally, a year has a total of 365 days which can also be written as 52 weeks and one day, (52 × 7) + 1. An extra day along with the 52 weeks, is called an odd day. Counting the number of odd days in a year depends on whether the year is a leap year or a non-leap year.

Similarly, if we talk about a century, i.e., 100 years, then a century has a total of 76 non-leap years and 24 leap years. Counting of odd days in a century shall be as follows:

24 non-leap 6 × 1 = 124 odd days.

## Counting the number of odd days in a year

Since an ordinary or non-leap year has 365 days in total, it has one odd day. But if a leap has 366 days, then the number of odd days increases by one, so a leap year has a total of two odd days.

When you divide 365 by 7, you get 1 as a residual in an average year of 365 days. A non-leap year, therefore, has one odd day.

When we divide 366 by 7, we get 2 as the remainder, indicating that a leap year has 2 odd days.

If you’re asked to count the number of odd days from the year 2007, what will be the answer? If you can calculate now itself, then well and good, otherwise we’ll be solving the same question once we’ve discussed other details too.

## Counting of odd days in a century

A century (100 years) comprises 76 non-leap years and 24 leap years. So, the number of odd days in a century shall account for 124 days. The detailed calculation is given as:

76 × 1 + 24 × 2 = 124 odd days = 17 weeks + days = 5 odd days in a century.

Counting of odd days in 200 years = 5 × 2, which will give an equal of 3 odd days in 200 years.

Counting of odd days in 400 years = 5 × 4 + 1, which will give 0 odd days in 400 years.

Similarly, odd days are related to the week of the days as well.

Days of the week |
No. Of odd days |
---|---|

Sunday |
0 |

Monday |
1 |

Tuesday |
2 |

Wednesday |
3 |

Thursday |
4 |

Friday |
5 |

Saturday |
6 |

Number of odd days in a month

Months |
Days |
Odd days |
---|---|---|

Jan, Mar, May, July, Aug, Oct, Dec |
31 |
3 |

Feb (Non-leap year) |
28 |
0 |

Feb (Leap year) |
29 |
1 |

Apr, Jun, Sept, Nov |
30 |
2 |

## Important points to Remember

counting the number of odd days in a year can become quicker and easier by following some shortcut tricks given below.

- When counting odd days equals zero between any two months in one year, the day if the dates will also happen to be the same.

Non-leap year |
Leap year |
---|---|

Jan- Oct |
Jan- Apr |

Feb- Mar |
Jan- July |

Feb- Nov |
Feb- Aug |

Mar- Nov |
Mar- Nov |

Apr-July |
Apr- July |

Sep-Dec |
Sep-Dec |

- The calendar tends to repeat itself after every 400 years. For example, if it happens to be a Monday on 3 Aug 2003, then it will be Monday on 3 Aug 2403 as well as 3 Aug 2803.
- When we talk about the first day of a century, then it must either be Monday, Wednesday, Friday or Saturday. Similarly, while considering the last day of a century, it can never be one of the following days, Tuesday, Thursday, or Sunday.

- There are zero odd days in 400 years, there is a single odd day in 300 years, then a total number of odd days in 200 years equals 3, and there are a total of 5 odd days in a century or 100 years.

While solving problems related to counting the number of odd days in a year, you can easily check and eliminate the wrong options by using the above shirt tricks. It will save you a lot of time that can be used in lengthier questions that involve longer solutions.

Let’s summarise all that we have understood so far by solving an example.