Finding remainder of a factorial-like product utilizing factor decomposition

To determine divisibility, find the prime factorization of the denominator 1261. Test for prime factors: it is not divisible by 2, 3, 5, or 11. Testing 13: 1261 \div 13 = 97. Thus, 1261 = 13 × 97.

Now, inspect the terms in the continuous numerator product: 91 × 92 \dots × 99. Notice that 91 is a multiple of 13 (91 = 13 × 7). Also, 97 is explicitly present as a term in the series.

Since the product contains both 13 × 7 and 97, it can be rewritten as: (13 × 97) × 7 × Other Terms = 1261 × 7 × Other Terms. Because 1261 is a direct internal factor of the massive product, the entire expression divides perfectly with no remainder.

Answer: (d).