Question

Assertion :The exponent of prime 5 in 234! is 56. Reason: If n is a positive integer and p is a prime number, then the exponent of P in n! is given by $\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\left[\frac{n}{p^3}\right]+......$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R} is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

We can write $\left[\frac{234}{5}\right]+\left[\frac{234}{25}\right]+\left[\frac{234}{125}\right]$
$=\left[46.8\right]+\left[9.36\right]+\left[1.872\right]$
$=46+9+1$
$=56$
Hence exponent of $5$ in $234!$ is $56$.

Therefore both assertion and reason are correct and reason is correct explanation for assertion.