# STEP III/2015/5

In the following argument to show that √2 is irrational, give proofs appropriate for steps 3, 5 and 6.

1. Assume that √2 is rational.

2. Define the set S to be the set of positive integers with the following property:

n is in S if and only if n√2 is an integer.

3. Show that the set S contains at least one positive integer.

4. Define the integer k to be the smallest positive integer in S

5. Show that (√2-1)k is in S.

6. Show that steps 4 and 5 are contradictory and hence that √2 is irrational.

Prove that $2^\frac{1}{3}$ is rational if and only if $2^\frac{2}{3}$ is rational.

Use an argument similar to that of part (i) to prove that $2^\frac{1}{3}$ and $2^\frac{2}{3}$ are irrational.