Is there a general function to derive the minimum number which can be divided by 1, 2, 3, 4 ...? 1 can be divided by 1
2 can be divided by 1, 2
6 can be divided by 1, 2, 3
12 can be divided by 1, 2, 3, 4
60 can be divided by 1, 2, 3, 4, 5
60 can be divided by 1, 2, 3, 4, 5, 6
420 can be divided by 1, 2, 3, 4, 5, 6, 7
840 can be divided by 1, 2, 3, 4, 5, 6, 7, 8

I just wonder whether there is a general function to derive this number.
Thanks.
 A: The question rephrased:
How does one determine $m_k$, the smallest number which is divisible by all the positive integers from 1 to $k$?
As a product of primes
Well, if one starts with the product $\tilde m_k = 1 \cdot 2 \cdot \dots \cdot k$, this is divisble by $1, 2, \dots, k$, but it's not the smallest. For example $1 \cdot 2 \cdot 3 \cdot 4 = 24$, but the smallest is 12, since 4 is divisble by 2. If we remove the 2 we get the smallest number. This suggest a general procedure.  
Take all the prime numbers $p_1, \dots, p_l$ from $\{1, 2, \dots, k\}$ and let $q_i$ be the largest integer such that $p_i^{q_i} \leq k$ and set  
$$m_k = p_1^{q_1} p_2^{q_2} \cdots p_l^{q_l}$$
which should be the smallest number which is divisble by $1, \dots, k$.
Recursively
Calculate $m_k$, let $m_{k-1}$ be given. 


*

*Is $m_{k-1}$ divisble by $k$? If so, set
$m_k = m_{k-1}$.

*Otherwise, set $m_k =
   \operatorname{lcm}(m_{k-1}, k)$,
where $\operatorname{lcm}$ is the
least common multiple.


This gives a nice implementation for computer using the following formula:
$$\operatorname{lcm}(a,b) = \frac{|a b|}{\operatorname{gcd}(a,b)}$$
Least common multiple
Of course, you can just set:
$$m_k = \operatorname{lcm}(2, 3, \dots, k)$$.
