Odd Days — The Engine of Calendar Questions

🎯 Learning Objective

Master odd days: remainders after dividing by 7, and the odd days of years and centuries.

💡 Concept

• Odd days = days left over after complete weeks: N days → N mod 7 odd days
• Ordinary year = 365 days = 52 weeks + 1 → 1 odd day
• Leap year = 366 days → 2 odd days
• Century blocks: 100 years → 5 odd days, 200 → 3, 300 → 1, 400 → 0
• Every 400 years the odd days reset to 0 — that is why the calendar runs in 400-year cycles

🧮 Key Formulas

Odd days = N mod 7

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Ordinary year → 1 odd day; Leap year → 2

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100 yr → 5, 200 yr → 3, 300 yr → 1, 400 yr → 0

✏️ Easy Example

Q. How many odd days are there in 250 days?

1. 250 = 7 × 35 + 5
2. Remainder 5 → 5 odd days

Answer: 5 odd days

🇮🇳 Real-Life Example

Your birthday moves one weekday ahead every ordinary year — Wednesday this year, Thursday next. That single step IS the odd day.

📝 Exam-Level Example

Q. Find the number of odd days in 300 years.

1. 100 years → 5 odd days
2. 300 years → 3 × 5 = 15
3. 15 mod 7 = 1

Answer: 1 odd day

📝 Exam-Level Example

Q. 1 January 2025 was a Wednesday. What day was 1 January 2026?

1. 2025 is an ordinary year → 1 odd day
2. Wednesday + 1 = Thursday

Answer: Thursday

🪄 Memory Trick

Memorise 5-3-1-0 for 100-200-300-400 years. Any big stretch of years breaks into 400-year blocks (0 odd days) + centuries + leftover years.

⚠️ Common Mistakes

• ❌ Computing 100 years as plain 100 ÷ 7 — the leap years inside make it 5 odd days
• ❌ Giving a leap year only 1 odd day
• ❌ Reporting odd days bigger than 6 — always reduce mod 7

🏆 Exam Tips

• ✅ Weekday shift = odd days: +1 for an ordinary year, +2 for a leap year
• ✅ While adding odd days, reduce with mod 7 at every step

📌 Summary

• Odd days = remainder after ÷ 7
• 365 → 1 odd day, 366 → 2
• Centuries: 5, 3, 1, 0
• Odd days push weekdays forward