Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 1 - Objective Problem 2

In fact, any positive integer can be written as a finite product of elements of $A$ (if we allow $1$ to be the empty product).

The proof proceeds by induction. The base case is $N=1,$ which is representable via the empty product; you can also use $2 = 1+\frac22,$ $3 = 1+\frac21,$ and $4 = (1+\frac22)(1+\frac22)$ as small examples if it helps.

Now suppose that every integer in $[1,2^a]$ is representable, where $a \ge 1.$ We will show that any integer in $(2^a,2^{a+1}]$ is representable. This is easy enough: suppose $2^a < N \le 2^{a+1}$ ; then $N = \left( 1 + \frac{2^a}{N-2^a} \right) (N-2^a),$ and $N-2^a$ is representable by the hypothesis, so $N$ is representable. So if every integer in $[1,2^a]$ is representable, then every integer in $[1,2^{a+1}]$ is representable. This shows by induction that every positive integer is representable, so the density is $\fbox{1}.$