Correct Option (1)

To determine the remainder when $85 \times 87 \times 89 \times 91 \times 95 \times 96$ is divided by 100, we analyze the prime factorization of both the divisor and the dividend.

The divisor is $100$ , which can be factorized as $100 = 2^{2} \times 5^{2} = 4 \times 25$ .

Next, we examine the prime factors present in the terms of the product $85 \times 87 \times 89 \times 91 \times 95 \times 96$ :

The term $85$ contains a factor of $5$ ( $85 = 5 \times 17$ ).
The term $95$ contains another factor of $5$ ( $95 = 5 \times 19$ ).
The term $96$ contains a factor of $4$ ( $96 = 4 \times 24$ ).

Therefore, the product $85 \times 87 \times 89 \times 91 \times 95 \times 96$ can be expressed as:

$(5 \times 17) \times 87 \times 89 \times 91 \times (5 \times 19) \times (4 \times 24)$

Rearranging the terms to highlight the factors of 100:

$(5 \times 5 \times 4) \times (17 \times 87 \times 89 \times 91 \times 19 \times 24)$

This clearly shows that the product contains $5 \times 5 \times 4 = 100$ as a factor. Consequently, the entire product is a multiple of 100 and is perfectly divisible by 100.

When a number is perfectly divisible by another number, the remainder is 0.

Incorrect Options:

Options 2 (1), 3 (2), and 4 (4) are incorrect. A non-zero remainder would imply that the product $85 \times 87 \times 89 \times 91 \times 95 \times 96$ is not a perfect multiple of 100. However, as demonstrated, the product explicitly contains all the prime factors required for divisibility by 100 ( $2^{2}$ and $5^{2}$ ), confirming that it is a multiple of 100 and thus leaves a remainder of 0.