Let p be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisble by 3 to the power k.

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

Let p be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3 to the power k.

Discover how to find the largest integer k such that the product of the first 100 positive odd integers is divisible by 3^k. This problem involves concepts from number theory, prime factorization, and divisibility. Suitable for high school students in Grades 11-12.

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