Question

prove that if a and b are 2 positive integers then a square + b square is even but not divisible by 4

Answer 1

any positive integer is of the form 2q+1  where q is some integer
suppose a=2n+1 and b= 2m+1 where m and n are some integers
a^2+b^2= (2n+1)^2 + (2m+1)^2
4n^2 +1 +4n + 4m^2 + 1 +4m 
4(m^2+n^2+m+n)+2
a^2+b^2=4p+2 where p is equal to (m^2+n^2+m+n)
4p and 2 are even numbers ,therefore 4p+2 is also an even number
a^2+b^2 is even number and leaves remainder 2 when divided by 4 
a^2+b^2 is even but not divisible by 4

Answer 2

Let us take the 2 positive integers be a = 1 and b = 5.
Then,
a²+b² = 1²+5²
=1+25=26
26 is even.It is not divisible by 4