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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Permutation and Combination

NDA 2022 · Mathematics · Permutation and Combination

What is the maximum value of n such that 5^n divides (30! +35 !),...

What is the maximum value of nn such that 5n5^n divides (30! +35 !), where nn is a natural number?

Options

  1. A4
  2. B6
  3. C7
  4. D8

Correct answer

C. 7

Step-by-step solution

30!+35!=30!+35.34.33.32.31.30!30!(1+35.34.33.32.31)1+35!.34.33.32.31 is not divisible by 5\begin{aligned} & 30!+35!=30!+35.34 .33 .32 .31 .30! \\ & 30!(1+35.34 .33 .32 .31) \\ & 1+35!.34 .33 .32 .31 \text{ is not divisible by } 5\end{aligned} So, maximum value of nn such that 5n5^n divides 30 ! =[305]+[3052]+[3053]+…=6+1+0+0…=7\begin{aligned} & =\left[\frac{30}{5}\right]+\left[\frac{30}{5^2}\right]+\left[\frac{30}{5^3}\right]+\ldots \\ & =6+1+0+0 \ldots=7\end{aligned}
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