Comprehensive theory, odd days calculations, century distributions, and authentic CBSE model solutions (2026-27).
A clock apprises us about smaller spans of time (seconds, minutes, hours), whereas a calendar depicts the days, weeks, and months of a specific year.
| Parameter | Ordinary Year | Leap Year |
|---|---|---|
| Number of Days | 365 Days | 366 Days |
| Number of Weeks | 52 Weeks and 1 Day | 52 Weeks and 2 Days |
| Number of Odd Days | 1 Odd Day ($365 = 52 \times 7 + 1$) | 2 Odd Days ($366 = 52 \times 7 + 2$) |
Since every 7 consecutive days form a complete week, the odd days in each calendar month are determined by the remainder of its days divided by 7:
| Month | Total Days | Odd Days ($Days \pmod 7$) |
|---|---|---|
| January | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| February | 28 / 29 days | $28 = 4\text{ wks} + 0 \implies \mathbf{0}$ (ordinary) $29 = 4\text{ wks} + 1 \implies \mathbf{1}$ (leap) |
| March | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| April | 30 days | $\frac{30}{7} = 4\text{ wks and } 2\text{ days} \implies \mathbf{2}$ |
| May | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| June | 30 days | $30 = 4\text{ wks} + 2 \implies \mathbf{2}$ |
| Month | Total Days | Odd Days ($Days \pmod 7$) |
|---|---|---|
| July | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| August | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| September | 30 days | $30 = 4\text{ wks} + 2 \implies \mathbf{2}$ |
| October | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
| November | 30 days | $30 = 4\text{ wks} + 2 \implies \mathbf{2}$ |
| December | 31 days | $31 = 4\text{ wks} + 3 \implies \mathbf{3}$ |
Coding and decoding is foundational to modern computing and cryptography (e.g. phone passwords, bank debit cards, hashing). In calendar mathematics, decoding a date into a day of the week is performed via Modulo 7 arithmetic.
A normal year is a leap year if it is divisible by 4. Fun Fact: A number is divisible by 4 if its last two digits are divisible by 4. However, for century years (100, 200, 300, 400, 1900, 2000), it must be divisible by 400 to be a leap year.
| Century Year | Calculation of Odd Days | Net Odd Days | Last Day of Century |
|---|---|---|---|
| 100 | 100 years | 5 | Friday |
| 200 | $5 + 5 = 10 = (7 \times 1) + 3$ | 3 | Wednesday |
| 300 | $5 \times 3 = 15 = (7 \times 2) + 1$ | 1 | Monday |
| 400 | $(5 \times 4) + 1 = 21$ (multiple of 7) | 0 | Sunday |
| 500 | $400 + 100 \implies 0 + 5$ | 5 | Friday |
| 600 | $400 + 200 \implies 0 + 3$ | 3 | Wednesday |
| 700 | $400 + 300 \implies 0 + 1$ | 1 | Monday |
| 800 | $2 \times 400 \implies 0$ | 0 | Sunday |
| 1000 | $800 + 200 \implies 0 + 3$ | 3 | Wednesday |
| 1400 | $1200 + 200 \implies 0 + 3$ | 3 | Wednesday |
| 1600 | Multiple of $400 \implies 0$ | 0 | Sunday |
| 1700 | $1600 + 100 \implies 0 + 5$ | 5 | Friday |
| 2000 | Multiple of $400 \implies 0$ | 0 | Sunday |
| 2100 | $2000 + 100 \implies 0 + 5$ | 5 | Friday |