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JEE Main 202624 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
24 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Permutation and Combination

JEE Main 2026 · 24 January · Evening Shift · Mathematics · Permutation and Combination

The largest value of n , for which 40^ n divides 60! , is

The largest value of nn, for which 40n40^{n} divides 60!60!, is

Options

  1. A11
  2. B12
  3. C14
  4. D13

Correct answer

C. 14

Step-by-step solution

40n=23n×5n40^n = 2^{3n} \times 5^n. We need the highest power of 4040 dividing 60!60!.
Power of 55 in 60!=⌊60/5⌋+⌊60/25⌋=12+2=1460! = \lfloor 60/5 \rfloor + \lfloor 60/25 \rfloor = 12 + 2 = 14.
Power of 22 in 60!=⌊60/2⌋+⌊60/4⌋+⌊60/8⌋+⌊60/16⌋+⌊60/32⌋=30+15+7+3+1=5660! = \lfloor 60/2 \rfloor + \lfloor 60/4 \rfloor + \lfloor 60/8 \rfloor + \lfloor 60/16 \rfloor + \lfloor 60/32 \rfloor = 30 + 15 + 7 + 3 + 1 = 56.
We need n≤14n \leq 14 and 3n≤563n \leq 56 (i.e., n≤18n \leq 18).
The binding constraint is n≤14n \leq 14, so the largest value is n=14n = 14.
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