How can you determine the maximum value of k such that 3^k is a factor of the product of integers from 50 to 150 that leave a remainder of 2 when divided by 4?

To determine the maximum value of ( k ) such that ( 3^k ) is a factor of the product of integers from 50 to 150 that leave a remainder of 2 when divided by 4, we need to first identify the specific integers in that range. The integers of concern are those of the form ( 4n + 2 ), where ( n ) is an integer, which results in even integers.

The correct choice emphasizes focusing on even multiples of 3 among these integers. This is essential because only even integers contribute to the factorial prime count when considering the factorization of the product. When calculating how many times 3 divides into this product, identifying which numbers are even and how many of them are multiples of 3 will give the correct count.

In contrast, simply counting all multiples of 3 from 50 to 150 does not account for the constraint of leaving a remainder of 2 when divided by 4. This would lead to including odd integers as well, which are irrelevant to the specific product we are interested in.

The approach of identifying just even multiples of 3 helps us directly focus on the integers we need, allowing us to correctly calculate the maximum exponent ( k \