Question

Find the least number by which \[162\] must be divided so that the number obtained is a perfect square. Also, find the square root of the resulting number.

Answer 1

Hint: In this question, we need to find the least number by which \[162\] must be divided so that the number obtained is a perfect square and also we need to find the square root of the resulting number. First, we can find the factors of the number \[162\]. Then we need to make the number \[162\] as a perfect square number. In order to make the number a perfect square number, we need to divide the number by a factor with no pair. Then on simplifying, we will get the resulting number. Then on taking the square root of the resulting number, we will get our required answer.

Complete step by step answer:
Given \[162\], first let us find the factors of the number \[162\] ,
\[162 = 2 \times 3 \times 3 \times 3 \times 3\] which can be written as \[162 = 2 \times 3^{4}\]. The number \[2\] in the factors of \[162\] is without a pair. So, if we divide \[162\] by \[2\], we get a perfect square. On dividing both sides by \[2\] we get,
\[162 \div 2 = 3^{4}\]
On simplifying we get,
\[\Rightarrow \ 162 \div 2 = 81 = 3^{4}\]
Thus \[2\] is the least number

Here we need to find the square root of the resulting number. That we need to find the square root of the number \[81\]. On taking square root on both sides we get,
\[\Rightarrow \ \sqrt{81} = \sqrt{3^{4}}\]
On taking the numbers out of the radical sign we get,
\[\Rightarrow \ 3^{2}\]
That is \[3 \times 3\] which is equal to \[9\]. Thus the square root of the number \[81\] is \[9\] .

Therefore, the least number by which \[162\] must be divided so that the number obtained is a perfect square is \[2\] and also its square root of \[81\] is \[9\].

Note: The concept used to solve these types of questions is the prime factorisation method . We can also use a calculator to simplify the final step. If we are using the calculator to find out the answer , we can simply enter the square root of \[81\] which will give us the correct answer. We should be very careful while taking the numbers out of the radical sign.