5. Among the $n$ positive integers from 1 to $n$, those with the most positive divisors are called the "prosperous numbers" among these $n$ positive integers. For example, among the positive integers from 1 to 20, the numbers with the most positive divisors are $12, 18, 20$, so $12, 18, 20$ are all prosperous numbers among the positive integers from 1 to 20. Then, the least common multiple of all the prosperous numbers among the positive integers from 1 to 100 is $\qquad$
5. 10080.

First, in the prime factorization of the first 100 positive integers, the maximum number of different prime factors is three. This is because the product of the smallest four primes is $2 \times 3 \times 5 \times 7=210$, which exceeds 100.

Second, to maximize the number of divisors, the prime factors should be as small as possible. The number with the maximum number of positive divisors can be determined through trial and classification enumeration, and the number with the maximum number of positive divisors is 12:

For numbers with only one prime factor, the one with the most positive divisors is $2^{6}=64$, which has seven positive divisors, less than 12;

For numbers with two different prime factors, those with 12 positive divisors are $2^{3} \times 3^{2}$ and $2^{5} \times 3$;

For numbers with three different prime factors, those with 12 positive divisors are
$$
2^{2} \times 3 \times 5, 2 \times 3^{2} \times 5, 2^{2} \times 3 \times 7 \text{. }
$$

It can be seen that among the first 100 positive integers, the numbers with 12 positive divisors are only five, and their least common multiple is
$$
2^{5} \times 3^{2} \times 5 \times 7=10080 \text{. }
$$