Question

The Square of an odd integer must be of the form
A.$6n + 1$
B.$6n + 3$
C.$8n + 1$
D.$4n + 1$ but may not be $8n + 1$

Answer 1

Hint: In mathematics a square number for perfect square is an integer that is the square of an integer. It is the product of some other integer with itself, for example $\sqrt{9}=3$ so $9$ is a perfect square number.

Complete step-by-step answer:
Given: The Square of an odd integer.
Square of any odd number can be written in the form of $4n+1$,
Like,
For $3,\,{{3}^{2}}=9$ but we can write $9$ in the form of $6n+3$ where $n$ is an integer. But we can write it as $4\times 2+1$ which is nothing but of the form $4n+1$. Similarly for $5,\,\,{{5}^{2}}=25$ but we can't write it in the form of $6n+1$ but on the other hand we can write it in the form of $6\times 4+1$ which is also of the form $4n+1$.
Similarly, a bigger square suppose ${{21}^{2}}=441$ and we can't write it in the form of $8n+1$ but it can be easily written as $4\times 110+1$ which is in the form of $4n+1$.
Thus, from all the above examples we can calculate that the square of any odd integer must be of the form $4n+1$
Hence the correct answer is option $\left( D \right)$
So, the correct answer is “Option D”.

Note: In this type of questions the biggest mistake made by the majority of students are they misunderstood the question statement and think it to be the square of number like ${{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}}....$ But it is not like that it's just the square of individual number example ${{3}^{2}}=9$ or
${{5}^{2}}=25$. Do not repeat these types of silly mistakes and think logically to get the answer.