Number systems - exercises

Exercise.

Prove that 2 is not rational, as follows. Suppose otherwise and try to get a contradiction. So suppose that 2=(

)2 for some integers

,

. Let be least such that there are

,

with 2=(

)2. Observe that 2.2=

2

and use this to deduce

,

are both even and hence there are smaller

','

with 2=(

'

' )2.

Exercise.

Use the idea of the proof that 2 is not rational to show that, for any positive integer , either is an integer or else is not rational.