Question

The square of an odd number is

• A. always an even number
• B. sometimes even and sometimes odd
• C. always an odd number
• D. always an irrational number

Answer 1

The correct option is C always an odd number

By definition, an odd number is an integer that can be written in the form $2n+1$, for some integer $n$, as an odd number leaves remainder 1 when divided by 2.

So square of an odd number,

$(2n+1{)}^2$

$=4n^2+4n+1$

$=2(2n^2+2n)+1$

$=2\left(k\right)+1$, where $k=2n^2+2n$ is an integer

Hence we have shown that the square of any odd number is odd.