Proving the Multiplication of Even Integers is a Multiple of 4: A Simple Proof

[nastygoalie89]

Homework Statement

Use direct proof to prove "The product of any two even integers is a multiple of 4."

Homework Equations

definition of even is n=2k

The Attempt at a Solution

My proof is going in circles/getting nowhere.

So far I have (shortened): By definition even n=2k, n=2j for some integer k
2k(2j) = 4kj = 4(kj) kj is an integer because k and j are integers
and the product of two integers is an integer
Not sure where to take it from there or if I even set the proof up correctly!

 

[Mark44]

Homework Statement

Use direct proof to prove "The product of any two even integers is a multiple of 4."

Homework Equations

definition of even is n=2k

The Attempt at a Solution

My proof is going in circles/getting nowhere.

So far I have (shortened): By definition even n=2k, n=2j for some integer k
2k(2j) = 4kj = 4(kj) kj is an integer because k and j are integers
and the product of two integers is an integer
Not sure where to take it from there or if I even set the proof up correctly!

You have the gist of it, but you should use different letters for the two even integers, say m and n.

m = 2k, and n = 2j, for integers k an j
mk = (2k)(2j) = 4kj, which is obviously a multiple of 4.