Question

Prove that if x and y are both
odd integers then xsquare +ysquare is even
but not divisible by 4.​

Answer 1

Answer:

Eg. 3square + 5square = 34.

1. a square of any odd number is always odd( since when a odd numbervis multiplied by any odd nimber the result is odd
2. the sum of the two squares will be even since odd number added to odd number is always even
3. it is not necessarhy that the resulting number will be divisible by 4 since every even number is not divisibe by 4. Just as the example above

Answer 2

Let x = 2m+1 and y = 2m+3 are odd positive integers,for every positive integer m.

Then ,       x²+y² = (2m+1)²+(2m+3)²

                         = 4m²+1+4m+4m²+912m

                         = 8m²+16m+10 = even

                         = 2(4m²+8m+5) or 4(2m²+4m+2)+1

Hence. x²+y² is even for every positive integer m but not divisible by 4.