Question

The square of any positive odd integer for some integer $m$ is of the form

• A. $7m+1$
• B. $8m+1$
• C. $8m+3$
• D. $7m+2$

Answer 1

The correct option is B $8m+1$

We know that any positive odd integer $n$ is of the form $4q+1$ or $4q+3$ where $q$ is some integer.

When, $n=4q+1$
$n^2=(4q+1{)}^2$
$=16q^2+8q+1$
$=8q(2q+1)+1$
$=8m+1$Where $m=q(2q+1)$
$=>n^2$ is in the form $8m+1$

When$n=4q+3$
$n^2=(4q+3{)}^2$
$=16q^2+24q+9$
$=16q^2+24q+8+1$
$=8(2q^2+3q+1)+1$
$=8m+1$ Where $m=(2q^2+3q+1)$
$=>n^2$ is in the form $8m+1$

Hence,$n^2$ is in the form $8m+1$ if $n$ is an odd positive integer.