Question

Can a rectangular number also be a square number$?$

Answer 1

Hint:
Rectangular numbers are represented in the form of $m \times n$, where m represents number of rows and n represents number of columns and square numbers are represented in the form of $n \times n$ or ${n^2}$. So by comparing these we can say whether the rectangular number is a square number or not.

Complete step by step solution:
A rectangular number is a number which is the product of two consecutive numbers. A number is said to be rectangular whenever the number is a product of two natural numbers where the natural numbers should not be $1$ and the number itself.
The rectangular numbers are represented in terms of $m \times n$.
Where $m$ and $n$ are two different natural numbers.
For example: $6$- The number $6$ is a rectangular number because it is $2$ rows and $3$columns. The below figure shows the arrangement of a rectangular number.

A square number is a number which is a product of a number which is multiplied by the number itself. A Square number is represented in the form of $n \times n$ or ${n^2}$, where n is a natural number.
For example: $9$- The number $9$ is a square number because it is having $3$ rows and $3$ columns. The below figure shows the arrangement of a square number.

Now, by the definition of rectangular numbers and square numbers we can say the all the rectangular numbers cannot be a square number, for example if I take an example of number $6$ it can be arranged as $2 \times 3$ but it is not a square number as it is not able to represent in terms of $n \times n$.

Note:
Not all the rectangular numbers are square numbers but some rectangular numbers can be square numbers. When it comes to square numbers all the square numbers can be rectangular numbers.