Smallest Number

The smallest positive integer $n$ that can be divided by all the integers from 20 to 30 is the least common multiplier of integers from 20 to 30. That is $n = \text{lcm } (20, 21, 22, ... 30)$ . This is equivalent to finding the product of all largest prime factors (powers) of $\displaystyle \prod_{k=20}^{30} k$ . That is $n = 2^3 3^3 5^2 7^1 11^1 13^1 17^0 19^0 23^1 29^1 = \boxed{3605401800}$ .