Askiitians Tutor Team
Last Activity: 4 Months ago
A rectangular number is a number that can be arranged in a rectangle of rows and columns. Specifically, it can be expressed as the product of two positive integers (not necessarily equal), meaning it can be written as n = a × b, where a and b are integers, and a ≠ b.
A square number is a number that can be expressed as the product of an integer with itself, meaning it can be written as n = x², where x is an integer.
Now, the question asks if a rectangular number can also be a square number. This means we need to determine if a number that can be written as n = a × b (with a ≠ b) can also be written as n = x² (where x is an integer).
For a number to be both rectangular and square, it must be possible to express it as both a × b (with a ≠ b) and as x² (where x is an integer). This is only possible when a = b because for it to be a square, both sides of the rectangle must be equal in length, making the shape a square.
In other words, a rectangular number where a ≠ b cannot also be a square number, because a square number requires both sides to be equal, i.e., a = b.
Thus, the only case where a rectangular number is also a square number is when the rectangle becomes a square, i.e., when a = b. Therefore, a rectangular number can be a square number only when it is a perfect square.
Example: 36 is a rectangular number (6 × 6), and it is also a square number (6²).