Question

Give a direct proof for each of the following statements about the integers.

If x is even and y is odd, then x + y is odd.

Answer 1

See explanation below

Explanation:

If x is even, then we know that x = 2k for a k ≥1

If y is odd, then we know that y = 2j -1 for a j≥1

Therefore, x + y can be written similarly as:

`x+y`

`2k +(2j-1)\\2(k+j)-1`

And clearly, 2(k + j) -1 is odd.

Therefore, the sum x + y is an odd number.