Question

Which one of the following cannot be the square of a natural number?
A. 30976
B. 75625
C. 28561
D. 143642

Answer 1

Hint: To solve this question, we will prime factorise all the given options individually. If we get the prime numbers in pairs, then it means that it is the square of a natural number and if we do not get the prime numbers as pairs, then it is not considered as the square of natural numbers.

Complete step-by-step answer:
It is given in the question that we have to find which out of the given options cannot be the square of a natural number. To find that, we will prime factorise each of the given numbers individually. If we get the prime numbers in pairs, then we will consider them as squares of natural numbers and if we do not get them in pairs, then we will not consider them as square of natural numbers. So, let us consider each option individually.
Option A. 30976
We will factorise it as,
$\begin{align}
  & 2\left| \!{\underline {\,
  30976 \,}} \right. \\
 & 2\left| \!{\underline {\,
  15488 \,}} \right. \\
 & 2\left| \!{\underline {\,
  7744 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3872 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1936 \,}} \right. \\
 & 2\left| \!{\underline {\,
  968 \,}} \right. \\
 & 2\left| \!{\underline {\,
  484 \,}} \right. \\
 & 2\left| \!{\underline {\,
  242 \,}} \right. \\
 & 2\left| \!{\underline {\,
  121 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \text{ }1 \\
 & =\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{11\times 11} \\
 & =2\times 2\times 2\times 2\times 11 \\
 & =176 \\
\end{align}$
So, we have got all the prime numbers in pairs and we get 176 as the square root of 30976. Therefore, option A is the square of a natural number 176.
Option B. 75625.
We can factorise it as,
$\begin{align}
  & 5\left| \!{\underline {\,
  75625 \,}} \right. \\
 & 5\left| \!{\underline {\,
  15125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  3025 \,}} \right. \\
 & 5\left| \!{\underline {\,
  605 \,}} \right. \\
 & 11\left| \!{\underline {\,
  121 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \text{ }1 \\
 & =\overline{5\times 5}\times \overline{5\times 5}\times \overline{11\times 11} \\
 & =5\times 5\times 11 \\
 & =25\times 11 \\
 & =275 \\
\end{align}$
We can see that we have got the pairs of all the prime numbers and we get 275 as the square root of 75625. Therefore, option B is the square of a natural number 275.
Option C. 28561.
Let us factorise it as,
$\begin{align}
  & 13\left| \!{\underline {\,
  28561 \,}} \right. \\
 & 13\left| \!{\underline {\,
  2197 \,}} \right. \\
 & 13\left| \!{\underline {\,
  169 \,}} \right. \\
 & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & \text{ }1 \\
 & =\overline{13\times 13}\times \overline{13\times 13} \\
 & =13\times 13 \\
 & =169 \\
\end{align}$
So, we get the prime numbers in pairs here also, and 169 is the square root of the number 28561. Therefore, option C is also the square of a natural number 169.
Option D. 143642.
We will factorise it as,
$\begin{align}
  & \text{ }2\left| \!{\underline {\,
  143642 \,}} \right. \\
 & 71821\left| \!{\underline {\,
  71821 \,}} \right. \\
 & \text{ }1 \\
\end{align}$
$=2\times 71821$
We can see that it is clear from the above that the factors of the number 143642 are not in pairs. So, it means that it is not the square of a natural number.
Hence, option D. is the correct option.

Note: This type of question should be solved carefully as it involves many factors of a number and while calculating the pairs or the multiple, one should be careful not to miss out any of the factors. We can also use the long division method to find the square root of the given numbers and then choose the right option. But it might get difficult and consume a lot of time to compute for such big numbers.