Question

8. Prove that if x and y are both odd positive integers then x² +y²is even
but not divisible by 4.​

Answer 1

Answer:

x2+y2 is even but not divisible by 4

Step-by-step explanation:

ANSWER

We know that any odd positive integer is of the form 2q+1, where q is an integer. 

So, let x=2m+1 and y=2n+1, for some integers m and n. 

we have x2+y2

x2+y2=(2m+1)2+(2n+1)2

x2+y2=4m2+1+4m+4n2+1+4n=4m2+4n2+4m+4n+2

x2+y2=4(m2+n2)+4(m+n)+2=4{(m2+n2)+(m+n)}+2

x2+y2=4q+2, when q=(m2+n2)+(m+n)

x2+y2 is even and leaves remainder 2 when divided by 4. 

x2+y2 is even but not divisible by 4.

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