Question

Can a rectangular number also be a square number?

Answer 1

Hint: In the given question, we have been asked that if we have a number which is rectangular (a number that can be expressed as product of two terminating rational numbers) can also be a square number (a number that can be expressed as the product of two terminating rational numbers which are equal).

Complete step-by-step answer:
Let \[m\] and \[n\] be two terminating rational numbers (that is, each of the two numbers can be expressed in form of \[\dfrac{a}{{{{10}^x}}}\] and \[a\] is going to be an integer), which are not necessarily distinct, then,
Rectangular number is a number of the form \[m \times n\], and
Square number is a rectangular number with one special condition that \[m = n\], hence a square number is a number of the form \[{m^2}{\rm{ }}\,or\,{\rm{ }}{n^2}\].
Thus, it is clear from the definition that – Yes, a rectangular number can also be a square number, and a number which is square is also always a rectangular, but the inverse of it is not always true, as \[32 = 4 \times 8\] is a rectangular number but it is not a square number \[32 = {4^2} \times 2\].


Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contain the known and the unknown and pick the one which is the most suitable and the most effective for finding the answer of the given question. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order.