A Hamming number is a positive number which has no prime factor larger than $5$.
So the first few Hamming numbers are $1,2,3,4,5,6,8,9,10,12,15$.
There are $1105$ Hamming numbers not exceeding ${10}^8$.

We will call a positive number a generalised Hamming number of type $n$, if it has no prime factor larger than $n$.
Hence the Hamming numbers are the generalised Hamming numbers of type $5$.

How many generalised Hamming numbers of type $100$ are there which don't exceed ${10}^9$?