Question

For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also, find the square root of the square number so obtained: $180$

Answer 1

Step 1: Find prime factors of the given number

By the method of prime factorization, the factors of $180$ can be determined as follows,

So, $180=2\times 2\times 3\times 3\times 5$

$\Rightarrow 180=\overline{2\times 2}\times \overline{3\times 3}\times 5$

Here, it can be observed that the prime factors $2$ and $3$ form a pair, while $5$ does not form a pair.

Step 2: Determine the perfect square number and its square root

Since $5$ could not form a pair.

Therefore, a perfect square number can be obtained by multiplying $180$ by $5$.

So, the perfect square number $=180\times 5=900$

Now, the square root of the above perfect square number is,

$\sqrt{900}=\sqrt{2\times 2\times 3\times 3\times 5\times 5}$

$\Rightarrow \sqrt{900}=\sqrt{\overline{2\times 2}\times \overline{3\times 3}\times \overline{5\times 5}}$

$\Rightarrow \sqrt{900}=2\times 3\times 5$

$\Rightarrow \sqrt{900}=30$

Hence, the smallest whole number by which $180$ should be multiplied to get a perfect square is $5$ and the required perfect square number and its square root are $900$ and $30$ respectively.