We are given the number 675, which is not a perfect square. We need to find the smallest natural number that we can divide 675 by to obtain a perfect square.

1. Problem Description

We are given the number 675, which is not a perfect square. We need to find the smallest natural number that we can divide 675 by to obtain a perfect square.

2. Solution Steps

First, we find the prime factorization of

6

7

5. $675 = 3 \times 225 = 3 \times 3 \times 75 = 3 \times 3 \times 3 \times 25 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5^2$.

So, $675 = 3^3 \times 5^2$.

For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 675, the exponent of 3 is 3 (odd) and the exponent of 5 is 2 (even). To make the exponent of 3 even, we need to divide by

3. Then the exponent becomes

2. So, we divide 675 by 3 to get $675/3 = (3^3 \times 5^2)/3 = 3^2 \times 5^2 = (3 \times 5)^2 = 15^2 = 225$, which is a perfect square.

Thus, the smallest natural number we need to divide 675 by to make it a perfect square is

3.

3. Final Answer

3