The problem asks to find the smallest natural number that we need to multiply 243 with to make it a perfect square.

1. Problem Description

The problem asks to find the smallest natural number that we need to multiply 243 with to make it a perfect square.

2. Solution Steps

First, we need to find the prime factorization of

2

4

3. $243 = 3 \times 81 = 3 \times 3 \times 27 = 3 \times 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5$.

So, we have $243 = 3^5$.

To make $3^5$ a perfect square, we need an even power. The smallest even number greater than 5 is

6. Therefore, we need to multiply $3^5$ by $3^{6-5} = 3^1 = 3$.

Then, we have $243 \times 3 = 3^5 \times 3 = 3^6 = (3^3)^2 = 27^2 = 729$. Since $27^2 = 729$, 729 is a perfect square.

3. Final Answer

3