Question

How many factors of 1080 are perfect squares?

Answer 1

Hint: We can directly find out by dividing the number with the smallest prime number. First we will find out all the factors of $ 1080 $ by dividing the number with the smallest prime number. From there we will find out the perfect squares.

Complete step by step answer:
Moving ahead with the question in step wise manner;
As we need to find out the number of perfect squares present in the factors of $ 1080 $ . So as we know that by dividing the number with the smallest prime number we can find out the factors of the number. So we will find out the factors using the same method. And then out of them we will figure out which all factors are perfect squares. Which will be the answer.
So using the method in our question; we have a number $ 1080 $ of which we need to find out the factors. So factors of $ 1080 $ are:
 $ \begin{align}
  & 1080=1,2,3,4,5,6,8,9,10,12,15,18,20,24,27,30, \\
 & 36,40,45,60,72,90,108,120,135,180,216,270,360,540,1080 \\
\end{align} $
So these are the factors of $ 1080 $ . So from these we need to find out the perfect squares. So as we know, a perfect square is a number that can be expressed as the product of two equal integers. For example, 25 is a perfect square because it is the product of two equal integers, \[5\text{ }\times \text{ }5\text{ }=\text{ }25\]. So in the above factors perfect squares are $ 4 $ which is the perfect square of 2 similarly $ 9,36 $ are the perfect squares of all the factors of $ 1080 $ .
So by this we can say there are three perfect squares that are $ 4,9 $ and $ 36 $ .
Hence answer is 3, i.e. 3 factors of 1080 are perfect squares.

Note: Factor is a number or algebraic expression that divides another number or expression evenly, i.e. with no remainder. So we can find it by dividing the number with the smallest number, or by using the factor tree method.