If x and y are odd integers,then prove that x^2+y^2 is even but not divisible by 4.
Answers
Answered by
4
hello there !!
we know that every odd positive integer is of the form 2q+1 for some integer q
so, let x=2p+1 and y=2r+1 for some integers p and r .
=> x²+y²= (2p+1)²+(2r+1)²
=> x²+y²= 4(p²+r²)+4(p+r)+2
=> x²+y²= 4[(p²+r²)+(p+r)]+2
=> x²+y²= 4a+2 where a = [(p²+r²)+(p+r)]
x²+y² is even and leaves remainder 2 when divided by 4
we know that every odd positive integer is of the form 2q+1 for some integer q
so, let x=2p+1 and y=2r+1 for some integers p and r .
=> x²+y²= (2p+1)²+(2r+1)²
=> x²+y²= 4(p²+r²)+4(p+r)+2
=> x²+y²= 4[(p²+r²)+(p+r)]+2
=> x²+y²= 4a+2 where a = [(p²+r²)+(p+r)]
x²+y² is even and leaves remainder 2 when divided by 4
Similar questions