Answer to Question #142685 in Combinatorics | Number Theory for Christine Joy Sato

Question

Statement: “The product of an even number and any other number is even.”
Using direct proof method.

Answer 1

Suppose n is even integer. Then by the definition of even numbers, n = 2k for some integer k.

Suppose m is an integer.

Then by substitution we have $m\cdot n=m(2k)=2(mk)=2q$ for some integer $q=mk$. Therefore by the definition of even numbers the product $m\cdot n$ is an even number.

 This completes the proof.

Therefore, the product of an even number and any other number is even.