What is the remainder when 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99 is divided by 1261?

To find the remainder when 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 x 99 is divided by 1261, we first need to factorize the divisor, 1261.

1. Factorize the divisor: We can test for prime factors. 1261 is not divisible by 2, 3, 5. 1261 / 7 is not an integer. 1261 / 11 is not an integer. 1261 / 13 = 97. So, 1261 = 13 x 97. Both 13 and 97 are prime numbers.

2. Examine the product: The product is 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 x 99. We need to check if both 13 and 97 are factors within this product.

   • The number 91 is part of the product. We know that 91 = 7 x 13. Therefore, 13 is a factor of the given product.
   • The number 97 is explicitly present as one of the terms in the product. Therefore, 97 is also a factor of the given product.

3. Conclusion: Since the product contains both 13 and 97 as factors, and 13 and 97 are prime numbers (and thus coprime), their product (13 x 97 = 1261) must also be a factor of the given product. If a number is a factor of another number, it means the latter is perfectly divisible by the former, leaving a remainder of 0.

Therefore, the remainder when 91 x 92 x ... x 99 is divided by 1261 is 0.

The final answer is D) 0.