Square of an odd number

by Ancient Tree·

Is the square of an odd number necessarily an odd number ?

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Solution by Ancient Tree

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Yes. If $a$ is an odd number, by definition, it can be written as $a=2b+1$ where $b$ is an integer.
So its square is given by:
$a^{2}=(2b+1)^{2}=(2b)^{2}+2\times 2b+1^{2}=4b^{2}+4b+1$ and this can be written:
$a^{2}=2\times(2b^{2}+2b)+1.$ In other words, it is of the form $a^{2}=2c+1$ , so $a^{2}$ is an odd number.

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