15. Trailing Zeros in n! (Factorial)

Factorial n! me trailing zeros kaise nikalte hain: v5 method, Legendre link, trailing zeros in base 10, base power (10^k) divisibility, ending zeros in expressions, SSC CGL examples & practice with answers.

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1) Trailing zeros kya hote hain?

Trailing zeros = number ke end me aane wale continuous zeros. Example: 12000 me trailing zeros = 3.

2) Factorial me zeros kahan se aate hain?

Zero end me tab aata hai jab number me factor 10 ho. Aur 10 = 2 × 5. Factorial (n!) me 2 ka count bahut zyada hota hai, isliye limiting factor 5 hota hai.

So, trailing zeros in n! = number of 5s in n! = v₅(n!)

3) Direct formula (most important)

Trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ...

Reason: multiples of 5 give one 5, multiples of 25 give extra one 5, multiples of 125 give extra one 5, etc.

4) Examples (SSC CGL)

Example 1: Trailing zeros in 10!

⌊10/5⌋ = 2, ⌊10/25⌋ = 0 ⇒ zeros = 2

Example 2: Trailing zeros in 25!

⌊25/5⌋ + ⌊25/25⌋ = 5 + 1 = 6

Example 3: Trailing zeros in 100!

⌊100/5⌋ + ⌊100/25⌋ + ⌊100/125⌋ = 20 + 4 + 0 = 24

5) “n! is divisible by 10^k” type

Agar n! me trailing zeros = t, to n! divisible by 10^k tab hoga jab t ≥ k.

• Example: 100! me t=24 ⇒ 100! divisible by 10²⁰ (Yes), 10³⁰ (No)

6) Trailing zeros in (n!)^m

(n!)^m me factors repeat hote hain, so: zeros = m × zeros(n!)

• Example: zeros in (25!)³ = 3×6 = 18

7) Trailing zeros in product/division type (basic idea)

Product me zeros add up (v5 add), division me subtract (if divisible). SSC me common:

• zeros in (100! / 50!) = v5(100!) − v5(50!)
• zeros in (n! × 10^k) = zeros(n!) + k

Example 4: Trailing zeros in 100!/50!

• v5(100!) = 20+4 = 24
• v5(50!) = ⌊50/5⌋+⌊50/25⌋ = 10+2 = 12
• So zeros = 24 − 12 = 12

8) Tricky but common: trailing zeros in n!!? (not double factorial)

SSC me usually n! hi puchte hain. Agar kabhi “consecutive products” diye ho (like 1×2×...×n) wo n! hi hota hai.

9) Common traps

• Sirf ⌊n/5⌋ lena enough nahi (25,125 extra 5s dete hain).
• 2 count karne ki zaroorat nahi (factorial me 2 always more than 5).
• Trailing zeros means end ke zeros, beech ke zeros nahi.

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10) Practice (SSC CGL) + Answers

1. Find trailing zeros in 30!.
2. Find trailing zeros in 50!.
3. Find trailing zeros in 125!.
4. How many trailing zeros in (20!)²?
5. How many trailing zeros in 100!/25! ?
6. Maximum k such that 200! is divisible by 10^k?

Show Answers

1. ⌊30/5⌋+⌊30/25⌋ = 6 + 1 = 7
2. ⌊50/5⌋+⌊50/25⌋ = 10 + 2 = 12
3. ⌊125/5⌋+⌊125/25⌋+⌊125/125⌋ = 25 + 5 + 1 = 31
4. zeros(20!) = ⌊20/5⌋ = 4 ⇒ (20!)² zeros = 2×4 = 8
5. v5(100!)=24; v5(25!)=6 ⇒ zeros = 24−6 = 18
6. zeros(200!) = ⌊200/5⌋+⌊200/25⌋+⌊200/125⌋ = 40 + 8 + 1 = 49