Question

Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3 for any integer q.

Answer 1

Let a be any positive integer

By Euclid's division lemma

$$a=bm+r$$

$$a=5m+r$$ (when $$b=5$$)

So, r can be $$0,1,2,3,4$$

Case 1:

$$\therefore a=5m$$ (when $$r=0$$)

$$a^2=25m^2$$

$$a^2=5(5m^2)=5q$$

where $$q=5m^2$$

Case 2:

when $$r=1$$ (when $$r=1$$)

$$a=5m+1$$

$$a^2=(5m+1)^2=25m^2+100m+1$$

$$a^2=5(5m^2+2m)+1$$

$$=5q+1$$ where $$q=5m^2+2m$$

Similarly

Case 3:

$$a=5m+2$$

$$a^2=25m^2+20m+4$$

$$a^2=5(5m^2+4m)+4$$

=$$5q+4$$

where $$q=5m^2+4m$$

Case 4:

$$a=5m+3$$

$$a^2=25m^2+30m+9$$
$$=25m^2+30m+5+4$$

$$=5(5m^2+6m+1)+4$$

$$=5q+4$$
where $$q=5m^2+6m+1$$

Case 5:

$$a=5m+4$$
$$a^2=25m^2+40m+16=25m^2+40m+15+1$$

$$=5(5m^2+8m+3)+1$$
$$=5q+1$$ where $$q=5m^2+8m+3$$

So none are in the form 5q+2 or 5q+3.