Question

Prove that the square of any positive integer is of the form 4q or 4q+1 for some integer q.

Answer 1

1.Let positive integer a = 4m + r , By division algorithm we know here 0 ≤ r < 4 , So
When r = 0
a = 4m
Squaring both side , we get
a2 = ( 4m )2
a2 = 4 ( 4m​2 )
a2 = 4 q , where q = 4m2

When r = 1
a = 4m + 1
squaring both side , we get
a2 = ( 4m + 1 )2
a2 = 16m2 + 1 + 8m
a2 = 4 ( 4m2 + 2m ) + 1
a2 = 4q + 1 , where q = 4m2 + 2m

When r = 2
a = 4m + 2
Squaring both hand side , we get
a2 = ​( 4m + 2 )2
a2 = 16m2 + 4 + 16m
a2 = 4 ( 4m2 + 4m + 1 )
a2 = 4q , Where q = ​ 4m2 + 4m + 1

When r = 3
a = 4m + 3
Squaring both hand side , we get
a2 = ​( 4m + 3 )2
a2 = 16m2 + 9 + 24m
a2 = 16m2 + 24m ​ + 8 + 1
a2 = 4 ( 4m2 + 6m + 2 ) + 1
a2 = 4q + 1 , where q = 4m2 + 6m + 2
Hence
Square of any positive integer is in form of 4q or 4q + 1 , where q is any integer . ( Hence proved )