How to calculate $lcm(2,3,4,5,6,7,8,9,10)$ solely by mental arithmetic? Denote $lcm(a,b)$ the lowest common multiple of $a$ and $b.$

Question: How to calculate 
  $$lcm(2,3,4,5,6,7,8,9,10)$$
  solely by mental arithmetic?

I have a way to calculate, but it relies on writing on paper. 
We can calculate the lcm using recursively, that is, 
$$l = lcm( lcm(2,3),4,5,6,7,8,9,10 ) = lcm(lcm(6,4),5,6,7,8,9,10) = lcm(lcm(12,5),6,7,8,9,10)...$$
I know that my notations above are not correct but hopefully the idea can get through.

Since we have a list of consecutive integers starting from $2$, just pick  the largest powers of a prime:
$$\text{lcm}(2,3,4,5,6,7,8,9,10)=\text{lcm}(5,7,8,9)=5\cdot 7\cdot 8 \cdot 9=63\cdot 40=2520.$$
