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UPSC Prelims 2026 CSAT, question 19

Topic: Number system

Q19

If 10m×1000×n=7525×2532×327510^m \times 1000 \times n=75^{25} \times 25^{32} \times 32^{75}, where nn is not divisible by 10, then the value of mm is

  1. (a) 101
  2. (b) 111Correct answer
  3. (c) 121
  4. (d) 131

Solution.

Answer: (b) 111

  1. Factoring the right-hand side into primes gives 7525×2532×3275=(3×52)25×(52)32×(25)75=325×550+64×2375=325×5114×237575^{25} \times 25^{32} \times 32^{75} = (3 \times 5^2)^{25} \times (5^2)^{32} \times (2^5)^{75} = 3^{25} \times 5^{50 + 64} \times 2^{375} = 3^{25} \times 5^{114} \times 2^{375}.
  2. The highest power of 10=2×510 = 2 \times 5 dividing this product is min⁡(114,375)=114\min(114, 375) = 114.
  3. Since 10m×1000×n=10m+3×n10^m \times 1000 \times n = 10^{m+3} \times n and nn is not divisible by 1010, we have m+3=114m + 3 = 114, which gives m=111m = 111.