Question

$\huge\mathfrak\purple{Heya\:Mate}$❤

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

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Answer 1

Let n be an integer

n=bm+r

b= 5 r=0,1,2,3,4

n=5m

n=5m+1

n=5m+2

n=5m+3

n=5m+4

Let n=5m

n^2=(5m)^2

=25m^2=5(5m^2)

Let 5m^2 be q

=5q

Let n= 5m+1

n^2=(5m+1)^2

=25m^2+10m+1

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

Let 5m^2+2m be q

=5q+1

Let n= 5m+2

n^2=(5 m+2)^2

=25m^2+20 m+4

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

Let 5m^2+4m be q

=5q+4

Let n=5m+3

n^2=(5m+3)^2

=25m^2+30m+9

=25m^2+30m+5+4

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

Let 5m^2+6m+1=q

=5q+4

Let n=5m+4

n^2=(5m+4)^2

=25m^2+40m+16

=25m^2+40m+15+1

=5(5m^2+8m+3)+1

Let 5m^2+8m+3 be q

=5q+1

As we can see that square of any positive integer is not of the form 5q+2 or 5q+3