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

(a) 101
(b) 111 ✓ UPSC's answer
(c) 121
(d) 131

Why the answer is (b)

• Express all terms in the equation as products of prime factors 2 and 5: $10^m = 2^m 5^m$, $1000 = 10^3 = 2^3 5^3$, $75 = 3 \times 5^2$, $25 = 5^2$, and $32 = 2^5$.

• Calculate the total power of 5 on the right-hand side (RHS): From $75^{25}$, we get $(5^2)^{25} = 5^{50}$; from $25^{32}$, we get $(5^2)^{32} = 5^{64}$; $32^{75}$ has no factor of 5. Total power of 5 on RHS is $50 + 64 = 114$.

• Calculate the total power of 5 on the left-hand side (LHS): From $10^m$, we get $5^m$; from $1000$, we get $5^3$; since $n$ is not divisible by 10, it contains no factor of 5. Total power of 5 on LHS is $m + 3$.

• Equate the powers of 5 on both sides: $m + 3 = 114$.

• Solve for $m$: $m = 114 - 3 = 111$.

• Therefore, the value of $m$ is 111, which corresponds to option (b).

Why the other options are wrong

(a) 101: Option (a) 101 is incorrect because it results from an arithmetic error in summing the exponents of 5 on the RHS or subtracting the constant term incorrectly.
(c) 121: Option (c) 121 is incorrect because it likely results from adding the exponent of 2 from the term 32^75 to the calculation for 5, or a miscalculation of the total power of 5.
(d) 131: Option (d) 131 is incorrect because it does not match the calculated value of 111 derived from equating the powers of the prime factor 5.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.