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.