Factors & Trailing Zeros

🎯 Learning Objective

Count the factors of a number and the zeros at the end of a factorial.

💡 Concept

• Prime factorise: N = a^p × b^q → number of factors = (p+1)(q+1)
• Every 10 = 2 × 5, and 2s are plenty — so zeros in n! = number of 5s
• Zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
• Perfect squares have an ODD number of factors

🧮 Key Formulas

N = a^p·b^q → factors = (p+1)(q+1)

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Zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋…

✏️ Easy Example

Q. How many factors does 72 have?

1. 72 = 2³ × 3²
2. Factors = (3+1) × (2+1) = 4 × 3

Answer: 12 factors

🇮🇳 Real-Life Example

Arranging 72 chairs into equal rows: every factor pair (8×9, 6×12…) is a possible arrangement — that's why factor counting matters.

📝 Exam-Level Example

Q. How many zeros are at the end of 100! ?

1. ⌊100/5⌋ = 20
2. ⌊100/25⌋ = 4
3. ⌊100/125⌋ = 0
4. Total = 20 + 4 = 24

Answer: 24 zeros

🪄 Memory Trick

Keep dividing by 5 and add quotients: 100→20→4→0. Add: 24. Done in 10 seconds.

⚠️ Common Mistakes

• ❌ Counting 2s as well (only 5s limit the zeros)
• ❌ Forgetting the +1 in the factors formula

🏆 Exam Tips

• ✅ Perfect square ⇒ odd factor count — a favourite trap
• ✅ n! zero-count questions repeat every year in RRB

📌 Summary

• Factors = product of (power+1)s
• Zeros in n! → count 5s: n/5 + n/25 + …
• Perfect squares → odd number of factors