Question

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

Answer 1

Let n and m be any two positive numbers.

x is odd number i.e., x= 2n+1
y is odd number i.e., y= 2m+1

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

as 2 is not divisible by 4.

x2 + y2 is not divisible by 4.

Answer 2

Step-by-step explanation:

Since x and y are odd positive integers, so let x = 2q + 1 and y = 2p + 1 ,

•°• x² + y² = ( 2q + 1 )² + ( 2p + 1 )² .

= 4( q² + p² ) + 4( q + p ) + 2 .

= 4{( q² + p² + q + p )} + 2 .

= 4m + 2 , where m = q² + p² + q + p is an integer .

•°• x² + y² is even and leaves remainder 2, when divided by 4 that is not divisible by 4.

Hence, it is solved

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