यदि $10^m$ x 1000 x $n = 75^25$ x $25^32$ x $32^75$ है, जहाँ n, 10 से विभाज्य नहीं है, तो m का मान क्या है?

Correct Answer: (B) 111

Rewriting the equation: $10^m$$\times$ 10^3 $\times$ n = 10^{m+3} n = 75^{25} $\times$ 25^{32} $\times$ 32^{75}. Prime factorizing the right-hand side: $75^{25} = (3$$\times$ 5^2)^{25} = 3^{25} $\times$ 5^{50}, $25^{32} = (5^2)^{32} = 5^{64}$, and $32^{75} = (2^5)^{75} = 2^{375}$. Thus, $\text{RHS}=2^{375}\times 3^{25}\times 5^{114}$$\text{RHS}$ = 2^{375} $\times$ 3^{25} $\times$ 5^{114}. Because $n$ is not divisible by 10, the factor $10^{m+3}$ must contain all common pairs of prime factors 2 and 5. The maximum power of 10 dividing the RHS is $\min(375, 114) = 114$. Therefore, $m + 3 = 114$$\implies$ m = 111.