What approach can be used to find the number of 3's in the product of all integers between 2 and 300, considering divisibility by 9^y?

To determine the number of 3's in the product of all integers between 2 and 300, particularly with respect to divisibility by (9^y), counting pairs of 3's is the most effective approach. This is because (9) can be expressed as (3^2), so to find out how many times (9) divides into the product of these integers, it is necessary to know how many factors of (3) are present in that product.

To find the number of factors of (3), all instances of (3) must be counted across the integers in the specified range. This includes not only the multiples of (3) but also the multiples of higher powers of (3) (like (9), (27), etc.), since each higher power contributes additional factors of (3). For each integer (n), you can calculate the number of times (3) is a factor by determining the number of integers in the range that are multiples of (3), adding the contributions from multiples of (9) and from multiples of (27), and so on.

Counting the pairs of (3's) directly addresses the need