Correct Option
To determine the remainder, the divisor must first be factorized. The prime factorization of 1261 is 13 × 97.
The given product is 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99.
Upon examining the terms within the product:
- The number 91 is a multiple of 13, specifically 91 = 7 × 13.
- The number 97 is explicitly present as a factor in the product.
Since the product contains both 13 (derived from 91) and 97 as factors, the entire product is divisible by 13 × 97. Consequently, the product is perfectly divisible by 1261.
When a number is perfectly divisible by another, the remainder is 0.
Incorrect Options
Options 1 (3), 2 (2), and 3 (1) are incorrect. The presence of 91 (which is 7 × 13) and 97 within the multiplicands ensures that the entire product is a multiple of 13 × 97, which equals 1261. Therefore, the division of the product by 1261 yields no remainder, making any non-zero remainder incorrect.