What is the remainder when 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99 is divided by 1261?
- (a) 3
- (b) 2
- (c) 1
- (d) 0 ✓ UPSC's answer
Why the answer is (d)
• Factorize the divisor 1261 to identify its prime components: 1261 = 13 × 97.
• Examine the product 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99.
• Observe that the term 97 is explicitly present in the product.
• Since 97 is a factor of the product and 97 is a factor of the divisor 1261, the entire product is divisible by 97.
• Additionally, 91 is divisible by 13 (91 = 7 × 13), so the product is also divisible by 13.
• Because the product contains both prime factors 13 and 97, it is divisible by their product 1261, resulting in a remainder of 0.
Why the other options are wrong
- (a) 3
- The remainder is 0, not 3, because the product is exactly divisible by 1261.
- (b) 2
- The remainder is 0, not 2, because the product is exactly divisible by 1261.
- (c) 1
- The remainder is 0, not 1, because the product is exactly divisible by 1261.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.