Prove that the product of two even integers is divisible by four.
The Attempt at a Solution
If a·b where (a^b)ЄZ+, then 4|(a·b)
If 2|a ^ 2|b, then (a^b)≥2 v (a·b)=0
If 2|(2·n) then 4|[2·(2·n)] for nЄZ
∴ 4|(a·b) where (a^b)ЄZ+
Is my proof correct? Is my notation for a positive integer correct (Z+), or should I not use the + and instead state that a,b≥0 a,bЄK? I'm not sure if that really proves it or not, but it's the best I have at the moment.
I was given much shorter proofs after I constructed this, and am aware it can be much shorter, but I mainly want to know if this is CORRECT or not.