Question 23
(BLR) 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:
(i) The first operation is either addition or multiplication.
(ii) Thereafter, additions and multiplications are used alternately.
(iii) In each addition one can choose independently whether to add $2$ or $n$.
(iv) In each multiplication, one can choose independently whether to multiply by $2$ or by $n$.
A positive integer that cannot be so obtained is said to be unattainable.
(a) Prove that if $n\ge9$, there are infinitely many unattainable positive integers.
(b) Prove that if $n=3$, all positive integers except $7$ are attainable.
(i) The first operation is either addition or multiplication.
(ii) Thereafter, additions and multiplications are used alternately.
(iii) In each addition one can choose independently whether to add $2$ or $n$.
(iv) In each multiplication, one can choose independently whether to multiply by $2$ or by $n$.
A positive integer that cannot be so obtained is said to be unattainable.
(a) Prove that if $n\ge9$, there are infinitely many unattainable positive integers.
(b) Prove that if $n=3$, all positive integers except $7$ are attainable.