Question

Find the smallest number by which $3645$ must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.

Answer 1

Step 1: Use prime factorisation of given number

$\because 3645=\left(3\times 3\right)\times \left(3\times 3\right)\times 5\times 3$

To make it paired, it needs to be divided by $5\times 3=15$ so that remaining number is a perfect square

So,

$$\begin{gathered} \frac{3645}{15}=\frac{\left(3\times 3\right)\times \left(3\times 3\right)\times 5\times 3}{15} \\ =3^2\times 3^2 \end{gathered}$$

Step 2: Find the square root of the obtained expression.

So,

$\begin{array}{rcl}& \Rightarrow & \sqrt{3^2\times 3^2}=3\times 3\\ & =& 9\end{array}$.

Hence, the smallest number to be divided by is $15$ and the square root of the resulting number is $9$.