Set of Integers. How many does it contain? AMC 2003 Senior(Australia)

A set of positive integers has the properties that

• Every member in th set, apart form 1, is divisible by at least one of $2,3,$ or $5$.
• If the set contains $2n, 3n,$ or $5n$ for some integer $n$, then it contains all three and $n$ as well.

The set contains between $300$ and $400$ numbers. Exactly how many does it contain?

I started with ${1,2,3,5}$, and tried to add some more numbers: say when $n=2$, we can add $2times2=4, 3times2=6, 5times2=10$, it became ${1,2,3,5,4,6,10}$. Now we had $6=2times3$, which $n=3$, we should add $3times3=9$ and $5times3=15$. Also $10=2times5$, we need to add $3times5=15$ and $5times5=25$. Now we had set ${1,2,3,5,4,6,10,9,15,25}$. But this step is too slow, and I got lost finally.