3

I've been working on this question, but I am not sure if I am actually doing it right or wrong. So far, I have these steps down, about which I am not entirely sure.

Show that if $n$ is an integer and $n^3 + 5$ is odd then $n$ is even using the technique of proof by contraposition.

My work so far:

If $n^3 + 5$ is not odd, but even, then $n^3+5=2a$. Thus $n^3= 2a-5$, and $$n=\sqrt[3]{2a-5}$$

I can tell my work is far off from the answer, but I thought I'd show my work in case people thought I haven't tried at all yet... Can someone show me the correct way of doing it?

apnorton
  • 17,706
Belphegor
  • 1,268
  • 6
  • 27
  • 51
  • 3
    There is absolutely no reason this should be voted to be closed. This is an excellent question and the person asking the question has shown plenty of work. – abnry Aug 09 '14 at 17:24
  • @nayrb "Excellent" is a bit of an exaggeration. But it's definitely good. – Daniel Fischer Aug 09 '14 at 20:03

3 Answers3

1

You are proving the converse instead of the contrapositive. Instead, prove that if $n$ is odd, then $n^3+5$ is even.

fahrbach
  • 1,783
1

The contrapositive of a statement "If p, then q" is "If not q, then not p". Showing one implication is equivalent to showing the other implication holds. More formally, we can write this as $$(p \Rightarrow q) \Leftrightarrow (! q \Rightarrow !p),$$ but don't worry if that isn't entirely clear.

The statement you are asked to show is "if $n^3+5$ is odd, then $n$ is even". The contrapositive is thus "if $n$ is not even, then $n^3+5$ is not odd". This is simplified to "if $n$ is odd, then $n^3+5$ is even."

Well, $n=2m+1$ if odd, and so we have $$n^3+5 = (2m+1)^3+5 = 8m^3+12m^2+6m+1+5 = 2(4m^3+5m^2+3m+3)$$ which means it is even. Hence the contrapositive is true and hence the original statement is true.

abnry
  • 14,664
  • Ohh, it was the other way around.. Thank you for clarifying. – Belphegor Aug 09 '14 at 17:26
  • Yes! You swap the $p$s and $q$s. As mentioned elsewhere, you were trying to show the converse, which is "not p implies not q" (and which is equivalent to "q implies p" using the contrapositive). – abnry Aug 09 '14 at 17:27
1

Start in another way, assume that n is odd (and derive that $n^3+5$ is even)

supinf
  • 13,433