Question

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

Answer 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 = 4n, Where n= ​ 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 = 4n+ 1 , where n= 4m2 + 6m + 2

Hence

Square of any positive integer is in form of 4n or 4n + 1 , where q is any integer . ( Hence proved )