Question

Prove that if x is odd, then ${{x}^{2}}$ is also odd.

Answer 1

- Hint: We will be using the concept of number system to solve the question. We will be using a fact that the multiplication of two odd numbers is odd for this we will first expand ${{x}^{2}}$ as ${{x}^{2}}=x\times x$then we will apply this property to solve the question.

Complete step-by-step solution -

Now, we have been given that x is odd and we have to prove that ${{x}^{2}}$is also odd.
Now to prove that ${{x}^{2}}$is odd. We need to understand that the multiplication of two odd numbers is an odd number. For example, if we take two odd numbers 3 and 5 respectively, then we see that their multiplication is 15 and is odd this statement is true. In general also not for only 3 and 5.
Now, since we know that the multiplication of two odd number is also odd we can write ${{x}^{2}}$ as,
${{x}^{2}}=x\times x$
We now can see that ${{x}^{2}}$ can be written as product of x two times and since x is odd we can say that ${{x}^{2}}$ is multiplication of two odd numbers.
Therefore, it is proved that ${{x}^{2}}$ is also odd.

Note: To solve these types of questions one should know the concept of number system. It is important to know the concept of multiplication of two numbers with respect to the parity of the number whether the number is even or odd for this one should remember the following facts
\[\begin{array}{*{35}{l}}
   Even\text{ }+\text{ }Even\text{ }=\text{ }Even \\
   \begin{align}
  & Odd\text{ }+\text{ }Odd\text{ }=\text{ }Even \\
 & Odd\text{ }\times \text{ }Odd\text{ }=\ \ Odd\text{ } \\
\end{align} \\
\end{array}\]