There isn’t too much going in this question.Nonetheless,I just liked it for some bloody reason.
- Let
and
be positive integers where
.If
is prime,prove that
is even and n is of the form
where
Lemma:Factorization of
where
is an odd positive integer.
Simply use the factorization formula to prove this.
Proof
Since ,it is quite clear that
,this means that
is odd and so
is even
is even.
Now,to prove the second claim,assume that is not a power of 2.
This means that some odd prime
such that
ie
where k is a positive integer.
Here is an odd positive integer so by the Lemma,
This even accounts for the case when is prime ie
Here the factor ,so it contradicts the fact that
is prime.
This completes the proof.