Direct proof question, "The product of an even integer and an odd integer is even".

Question

I didnt understand 

Let say X = 2a +1 while y = 2b+1

 

Why you need to multiply xy rather x+y, as you can see below:

XY = (2a+1)(2y+1) <- correct

X+Y = 2a+1+2y+1 <- Incorrect

 

Can you elobrate why it multiply rather than add. Kinda lost. Thank you maám/sir.

Answer 1

All even integers are multiples of 2 and all multiples of 2 are even integers.

One more than any even integer will be an odd integer.

Note:  Product means multiply.

 

Even integer:  2x

Odd integer: 2x + 1

Product of even and odd integer:  2x(2x + 1) = 4x² + 2x = 2(2x² + x)

Since 2(2x² + x) is a multiple of 2, it will be even.  This means that the

product of an even integer and an odd integer will be even.