We are tasked with finding the largest integer $k$ such that the product of the first 100 positive odd integers is divisible by $3^k$.

Let $P$ denote the product of the first 100 odd integers:

$P = 1 \times 3 \times 5 \times \cdots \times 199$

To determine the largest integer $k$ such that $3^k$ divides $P$, we need to count the number of factors of 3 in the product $P$. Specifically, we are interested in how many times 3 appears as a factor in the numbers from 1 to 199.

Step 1: Identify multiples of