Question

Consider the statement: For all integers n, if n3 is even then n is even. a. Explicitly write out what you are supposing and what you are trying to prove, if you are proving the statement by proving the contrapositive. Suppose _______________________________________________________________________ Need to show ____________________________________________________________________ b. Prove the statement by the method of the contrapositive

Answer 1

Answer:

It is proved that if $n^3$ is even the n is even.

Step-by-step explanation:

Given n is any integer.

To show $n^3$ is even then n is even.

Proving by contrapositive suppose $n^3$ is odd. Then we need to show n is odd.

Then, letting k is a ny integer,

$n^3=2k+1\implies n=(2k+1)^{\frac{1}{3}}$

Now since (2k+1) is odd therefore n is odd.

Conversly let n is odd, then,

$n=2k+1\implies n^3=(2k+1)^3$

since 2k+1 is odd so $n^3$ is odd.

This proves, if n is even then $n^3$ is even.