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

Factor the RHS: 75^25 = (3×5²)^25 = 3^25 × 5^50. 25^32 = 5^64. 32^75 = (2^5)^75 = 2^375. So RHS = 2^375 × 3^25 × 5^(50+64) = 2^375 × 3^25 × 5^114. LHS: 10^m × 1000 × n = 2^m × 5^m × 2^3 × 5^3 × n = 2^(m+3) × 5^(m+3) × n. For n not divisible by 10, n contains no factor of 2 AND no factor of 5 simultaneously (i.e., not both). Matching powers: m+3 ≤ 114 (power of 5) and m+3 ≤ 375 (power of 2). To not have n divisible by 10, n shouldn't have both 2 and 5. The limiting factor is 5^114. Take m+3 = 114, so m = 111. Then 2^(m+3) = 2^114, but RHS has 2^375, so remaining 2^(375-114) = 2^261 goes into n. Then n = 2^261 × 3^25, which has factor 2 but no 5, so n is not divisible by 10 ✓. Hence m = 111.