Number Theory – Exponent of Prime in n! – JEE Main 2 Apr 2025 Shift 1

Solution:

$$E_p(n!) = \left[ \frac{n}{p} \right] + \left[ \frac{n}{p^2} \right] + \left[ \frac{n}{p^3} \right] + \dots$$

$$E_3(50!) = \left[ \frac{50}{3} \right] + \left[ \frac{50}{9} \right] + \left[ \frac{50}{27} \right] + \left[ \frac{50}{81} \right]$$

$$E_3(50!) = 16 + 5 + 1 + 0$$

$$E_3(50!) = 22$$

Ans. (2)