IMO Shortlist 1998 problem C2

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2. travnja 2012.
Let n be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties:

1.) The first operation is either addition or multiplication.

2.) Thereafter, additions and multiplications are used alternately.

3.) In each addition, one can choose independently whether to add 2 or n

4.) In each multiplication, one can choose independently whether to multiply by 2 or by n.

A positive integer which cannot be so obtained is said to be unattainable.

a.) Prove that if n\geq 9, there are infinitely many unattainable positive integers.

b.) Prove that if n=3, all positive integers except 7 are attainable.
Izvor: Međunarodna matematička olimpijada, shortlist 1998