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

The correct option is 0.

To find the remainder when the product of the numbers from 91 to 99 is divided by 1261, we can analyze the prime factorization of the divisor, 1261.
Let us first find the factors of 1261 by testing small prime numbers. Since 1261 is not divisible by 2, 3, 5, 7, or 11, we test the next prime number, 13:

$$1261÷13=97$$

Since 97 is itself a prime number, we can write the prime factorization of 1261 as:

$$1261=13\times 97$$

Now, let us examine the given product in the numerator:

$$P=91\times 92\times 93\times 94\times 95\times 96\times 97\times 98\times 99$$

We need to check if the factors of the divisor (13 and 97) are present within this product:
1. The first term in the product is 91, which is a multiple of 13 since $91=13\times 7$.
2. The seventh term in the product is 97, which is directly present as a factor.

Since the product contains both 13 (within 91) and 97 as factors, the entire product must be divisible by their product:

$$13\times 97=1261$$

Because the product is a multiple of 1261, it divides completely with no remainder. Therefore, the remainder is 0.