I'm brushing up for my Discrete Math final exam, and would like to know if the following proof is valid...
Prove by direct proof that $a^2-5a+8$ is even for any integer $a$.
Proof: By factoring, we achieve $a(a-5)+8$.
Suppose $a$ is an even integer. Then $a(a-5)$ would be even, as the product of an even integer and another integer is an even integer. So, $a(a-5)+8$ would also be even, as the sum of an even integer and an even integer is even.
Suppose $a$ is an odd integer. Then $a-5$ would be even, as the difference of two odd integers is even. So, $a(a-5)$ would also be even, as the product of an even integer and another integer is an even integer. So, $a(a-5)+8$ would also be even, as the sum of an even integer and an even integer is even.
Therefore, for any integer $a$, $a^2-5a+8$ will be even.
Let me know if this is a valid & satisfactory formal proof, or of anything I could change or improve. Thanks! Sam