Question

Show that the square of any positive integer cannot be of the form $5q+2\mathrm{or}5q+3$ for any integer $q$.

Answer 1

Proving that for any integer $q$, the square of a positive integer cannot be of the form $5q+2\mathrm{or}5q+3$.

According to Euclid's division algorithm, $a=bm+r$, where $a,b,m,r$ are non-negative integers and $0\le r<b$.

Assume $a$ be any positive integer and $b=5$.

Substitute $r=0,1,2,3,4$ and $b=5$ into $a=bm+r$.

We have

$\begin{array}{rcl}a& =& 5m\left[\therefore r=0\right]\\ a& =& 5m+1\left[\therefore r=1\right]\\ a& =& 5m+2\left[\therefore r=2\right]\\ a& =& 5m+3\left[\therefore r=3\right]\\ a& =& 5m+4\left[\therefore r=4\right]\end{array}$

Case A. Square $a=5m$ on both sides.

$\begin{array}{rcl}{a}^2& =& {\left(5m\right)}^2\\ & \Rightarrow & a^2=25m^2\\ & \Rightarrow & a^2=5\left(5m^2\right)\\ & \Rightarrow & a^2=5q\left[q\mathrm{is}\mathrm{some}\mathrm{integer}\mathrm{and}q=5m^2\right]\end{array}$

Case B. Square $a=5m+1$ on both sides.

$\begin{array}{rcl}{a}^2& =& {\left(5m+1\right)}^2\\ & \Rightarrow & a^2=25m^2+10m+1\\ & \Rightarrow & a^2=5\left(5m^2+2m\right)+1\\ & \Rightarrow & a^2=5q+1\left[q\mathrm{is}\mathrm{some}\mathrm{integer}\mathrm{and}q=5m^2+2m\right]\end{array}$

Case C. Square $a=5m+2$ on both sides.

$\begin{array}{rcl}{a}^2& =& {\left(5m+2\right)}^2\\ & \Rightarrow & a^2=25m^2+20m+4\\ & \Rightarrow & a^2=5\left(5m^2+4m\right)+4\\ & \Rightarrow & a^2=5q+4\left[q\mathrm{is}\mathrm{some}\mathrm{integer}\mathrm{and}q=5m^2+4m\right]\end{array}$

Case D. Square $a=5m+3$ on both sides.

$\begin{array}{rcl}{a}^2& =& {\left(5m+3\right)}^2\\ & \Rightarrow & a^2=25m^2+30m+9\\ & \Rightarrow & a^2=5\left(5m^2+6m+1\right)+4\\ & \Rightarrow & a^2=5q+4\left[q\mathrm{is}\mathrm{some}\mathrm{integer}\mathrm{and}q=5m^2+6m+1\right]\end{array}$

Case E. Square $a=5m+4$ on both sides.

$\begin{array}{rcl}{a}^2& =& {\left(5m+4\right)}^2\\ & \Rightarrow & a^2=25m^2+40m+16\\ & \Rightarrow & a^2=5\left(5m^2+8m+3\right)+1\\ & \Rightarrow & a^2=5q+1\left[q\mathrm{is}\mathrm{some}\mathrm{integer}\mathrm{and}q=5m^2+8m+3\right]\end{array}$

So square of no positive number can be of the form $5q+2\mathrm{or}5q+3$

Hence, it is proved that for any integer $q$, the square of a positive integer cannot be of the form $5q+2\mathrm{or}5q+3$.