Question

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

Answer 1

Answer.

Let the two odd positive numbers be x = 2k + 1 a nd y = 2p + 1 Hence x2 + y2 = (2k + 1)2 + (2p + 1)2 = 4k2 + 4k + 1 + 4p2 + 4p + 1 = 4k2 + 4p2 + 4k + 4p + 2 = 4(k2 + p2 + k + p) + 2 Clearly notice that the sum of square is even the number is not divisible by 4 Hence if x and y are odd positive integers, then x2 + y2 is even but not divisible by 4

Answer 2

answer: -

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.