Question

Which of the following numbers are perfect squares?
11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121

Answer 1

Hint: In this question, we have to find whether the given numbers are perfect squares. Therefore, we should find the greatest integer such that its square is smaller than all the given numbers, then we can take the next integers and check if their square is equal to any number in the given set. We should continue this process till the square of the integer exceeds all the numbers in the given set.

Complete step-by-step answer:
We know that the smallest number in the given set is 11. Now, we should find the greatest whole number whose square is closest to but less than 11. We find that as ${{3}^{2}}=9$ and ${{\left( 3+1 \right)}^{2}}={{4}^{2}}=16$ , is the required integer…………………(1.1)
Now, if any given number in the set is a perfect square, it should be the square of a number whose magnitude is greater than 3. Therefore, we can take numbers greater than 3 and check if their square is equal to any number in the given set. We find that
$\begin{align}
  & {{3}^{2}}=9 \\
 & {{4}^{2}}=16 \\
 & {{5}^{2}}=25 \\
 & {{6}^{2}}=36 \\
 & {{7}^{2}}=49 \\
 & {{8}^{2}}=64 \\
 & {{9}^{2}}=81 \\
 & {{10}^{2}}=100 \\
 & {{11}^{2}}=121.................(1.2) \\
\end{align}$
Now, as ${{11}^{2}}=121$ is the largest number in the given set, any of the given numbers cannot be the square of a number greater than 11. Therefore, those numbers in the set which match the RHS of any equation in (1.2) will be perfect squares and the others will not be perfect squares. Comparing (1.2) to the given numbers and using the above argument we find that
16, 36, 64, 81 and 121 are perfect squares and
11, 12, 32, 50, 79, 111 are not perfect squares which is the required answer.

Note:We could also have found out the square root of the given numbers and checked whether they are integers or not. Those numbers whose square root is an integer are perfect squares and those numbers whose square root is not an integer are not perfect squares. However, the answer obtained by this method would be the same as obtained in the solution.