Question

show that the square of any positive integer cannot of the form 5q+2or 5q+3 for any integer q​

Answer 1

the square of any positive integer cannot be in the form of 5q+2 it 5q+3 because

a=bq+r

r=0

5q+0

(5q)2

25q2

5×5q2/m

where 5q2 =m

5m

r=0

5q+1

(5q+1)2

(25q2+10q)+1

5(5q+2q)+1

-----------

m

where 5q2+1 =m

5m+1

r=0

5q+2

(5q+2)2

(25q2+20q)+4

5(5q+4q)+4

-----------

m

where 5q2+4 =m

5m+4

hence it cannot be in the form of 5q+2 or 5q+3