Question

Find the perfect square numbers between
(i) 30 and 40
(ii) 50 and 60

Answer 1

Hint: In this type of problem, you must remember the square of the number starting from 1 to 10. square of the number is equal to the product of the number with itself.
Example:

\[\begin{align}
  & {{a}^{2}}=a\times a \\
 & {{b}^{2}}=b\times b \\
\end{align}\]
Similarly
$\begin{align}
  & {{4}^{^{2}}}=4\times 4 \\
 & {{7}^{2}}=7\times 7 \\
\end{align}$

Complete step-by-step answer:
We know that square of ${{1}^{2}}=1$ and ${{10}^{2}}=100$, so we can say that square of numbers between 1 to 10 always lies between 1 to 100.
Now let’s find the square of numbers from 1 to 10, then try to visualize which perfect square number lies between 30 and 40, Similarly, we can also visualize which perfect square number lies between 50 and 60.

$\begin{align}
  & {{1}^{2}}=1 \\
 & {{2}^{2}}=4 \\
 & {{3}^{2}}=9 \\
 & {{4}^{2}}=16 \\
 & {{5}^{2}}=25 \\
 & {{6}^{2}}=36 \\
 & {{7}^{2}}=49 \\
 & {{8}^{2}}=64 \\
 & {{9}^{2}}=81 \\
 & {{10}^{2}}=100 \\
\end{align}$

(i) From the above calculation, we can easily say that 36 is a perfect square number between 30 and 40

(ii) From the above calculation, we can easily say that there is no such perfect square number that exists between 50 to 60.

Note: Another definition of perfect square number is, a number is said to be a perfect square number, if and only if the prime factor of a number is grouped in a pair of equal factors.
 Example: let’s take 144, now try to find the prime factor of 144

We get
$144=2\times 2\times 2\times 2\times 3\times 3$
      $=\left( 2\times 2 \right)\times \left( 2\times 2 \right)\times \left( 3\times 3 \right)$ (Grouping the factor into the pairs of equal factors)
      $\begin{align}
  & ={{2}^{2}}\times {{2}^{2}}\times {{3}^{2}} \\
 & ={{\left( 2\times 2\times 3 \right)}^{2}} \\
 & ={{12}^{2}} \\
\end{align}$

Therefore, 144 is a perfect square number..