What is the remainder when
\(91 \times 92 \times 93 \times 94 \times
95 \times 96 \times 97 \times 98 \times 99\)is divided by 1261?

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.