यदि 15 × 14 × 13 × ... × 3 × 2 × $1 = 3^m$ × n, जहाँ m और n धनात्मक पूर्णांक हैं, तो m का महत्तम मान क्या है?

Correct Answer: (B) 6

The product $15$$\times$ 14 $\times$ $\dots$ $\times$ 1 represents $15!$. By Legendre's formula, the highest power of prime 3 dividing $15!$ is given by $E_3(15!) = \lfloor$$\frac{15}{3}$ \rfloor + \lfloor $\frac{15}{9}$ \rfloor = 5 + 1 = 6. Alternatively, counting powers of 3 among multiples in the product: 3 (one), 6 (one), 9 (two), 12 (one), and 15 (one) gives $1 + 1 + 2 + 1 + 1 = 6$. Thus, the maximum value of $m$ is 6.