Question

Prove that, if x and y are both odd positive integers, then x2 + yz is even but not divisible by 4.​

Answer 1

Let the two odd numbers be

(2a+1) & (2b+1) because if we add 1 to any even no. it will be odd.

x²+y²

=>(2a+1)²+(2b+1)²

=>(4a²+4a+1)+(4b²+4b+1)

=>4(a²+b²+a+b)+2

4 Is not a multiple of 2 it means clearly that 4 is not multiple of x²+y² , so x²+y² is even but not divisible by 4.

Hence proved.

Please mark me as brainliest.

Answer 2

SOLUTION ☺️

▶️Any odd positive integer is of the form 2q+1, where q is some integer.

suppose x= 2n+1 and y= 2m+1, where m & n are some integer.

=) x^2+y^2= (2x+1)^2+(2m+1)^2

=) 4n^2+1+4n+4m^2+1+4m

=) 4(m^2+n^2+m+n)+2

=) x^2+y^2= 4p+2, where p= m^2+n^2+m+n

4p and 2 are even numbers= 4p+2 is an even no.

=) x^2+y^2 is even no. and leaves remainder 2 when divided by 4.

=) x^2+y^2 is even but not divisible by 4.

HOPE it helps ✔️☺️