CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Prove that, if x and y are odd positive integers, then x2+y2 is even but not divisible by 4.

Open in App
Solution

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.

flag
Suggest Corrections
thumbs-up
2
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Introduction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon