Question

show that the square of any positive integer is of the form 4q ,4q+1

Answer 1

Let a integer 'a' is in the form of 4m

so

0≤r<b
⇒r=0,1

Now
Case 1) r=0
a=4m+0
sq. both the sides
a²=16m²
a²=4(4m²)
a²=4q            where q=4m²

Case 2) r=1
a=4m+1
sq. both the sides
a²=(4m+1)²
a²=16m²+1+8m
a²=16m²+8m+1
a²=4(4m²+2m)+1
a²=4q+1           where q=4m²+2m