Introduction

When dealing with the problem of finding the smallest possible positive value of n such that both 5^2 and 3^2 are factors of a number x n cdot 2^5 cdot 6^2 cdot 7^3, it’s essential to understand prime factorization and how to combine factors effectively. This article will guide you through the process with a step-by-step explanation and provide a clear solution.

1. Understanding the Prime Factorization

Let's start with the prime factorization of the given term x n cdot 2^5 cdot 6^2 cdot 7^3. First, we need to break down the term 6^2 into its prime factors.

The prime factorization of 6 is:

6 2 cdot 3

Thus, 6^2 (2 cdot 3)^2 2^2 cdot 3^2.

2. Substituting and Simplifying

Substituting 6^2 back into the expression for x:

x n cdot 2^5 cdot 2^2 cdot 3^2 cdot 7^3

Combining the powers of 2 results in:

x n cdot 2^{5 2} cdot 3^2 cdot 7^3

x n cdot 2^7 cdot 3^2 cdot 7^3

3. Ensuring 52 and 32 Factors

Now, to ensure that 5^2 is a factor of x, we need to consider the term n. Since x does not currently include any factors of 5, the term n must include 5^2 to make x divisible by both 5^2 and 3^2.

Additionally, as 6^2 2^2 cdot 3^2 is already a factor of x, n does not need to contribute any additional factors of 3.

Therefore, the smallest possible value of n that includes 5^2 is:

n 5^2 25

Thus, the smallest possible positive value of n is boxed{25}.

Conclusion

To summarize, when both 5^2 and 3^2 need to be factors of a number x, it's important to factorize the given term, combine like factors, and ensure the necessary prime factors are included. This example illustrates the step-by-step process of finding such a value of n.