Prove or disprove the following proposition: If $n^3 − 5$ is an odd integer, then $n$ is even.
I know that $n$ must be even in order for $n^3 – 5$ to be odd which means I have to prove the statement.. possibly with a contradiction? I have been able to successfully start the proof but I am unsure of where to go from here. Any help would be greatly appreciated!
Proof: Suppose $n$ is an odd integer which can be expressed as $n=2k+1$, and $n^3-5$ is also an odd integer.
$$(2k+1)^3 – 5 = (2k+1)(2k+1)(2k+1) – 5 = 8k^3 + 12k^2 + 6k + 1 – 5…$$
Where should I go from here? Thanks!
Best Answer
You could say if $n=2k+1$ then $n^3-5=8k^3+12k^2+6k-4$ is even (being a sum of even numbers), which proves it by contrapositive, but I find it simpler to say if $n^3-5$ is odd, then $n^3$ is even, so $n$ is even.