If $10^m$ x 1000 x $n = 75^25$ x $25^32$ x $32^75$, where n is not divisible by 10, then the value of m is

Correct Option: (c) 121

We are given the equation: 10^m x 1000 x n = 75^25 x 25^32 x 32^75, where n is not divisible by 10. We need to find the value of m. Let's express both sides in terms of prime factors 2 and 5. Right side: 75^25 x 25^32 x 32^75. Note that 75 = 3 x 25 = 3 x 5^2. So 75^25 = 3^25 x (5^2)^25 = 3^25 x 5^50. Next, 25^32 = (5^2)^32 = 5^64. Next, 32^75 = (2^5)^75 = 2^375. Combining these on the right side: Prime factors of the right side include 2^375, 3^25, and 5^(50 + 64) = 5^114. Thus, right side = 2^375 x 3^25 x 5^114. Left side: 10^m x 1000 x n = 10^m x 10^3 x n = 10^(m+3) x n = 2^(m+3) x 5^(m+3) x n. Since n is not divisible by 10, n does not contain both 2 and 5 as factors. Comparing the powers of 5 on both sides: Left side has 5^(m+3) (along with whatever powers of 5 might be in n, but n is not divisible by 10 so n cannot contribute factors of 10, meaning n cannot contain both 2 and 5). Wait, the power of 5 on the right side is 114. Therefore, m + 3 must be 114, which gives m = 111. Let's also check the power of 2: Right side has 2^375. Left side has 2^(m+3) x n. Since m = 111, m + 3 = 114, leaving 375 - 114 = 261 powers of 2 inside n (which is not divisible by 10 since it has no factor of 5). Thus, m = 111.