2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

The smallest natural number $k$ that is divisible by natural numbers $a$ and $b$ is called their *least common multiple*; we'll denote this by $a \wedge b$.
The inverse notion, the *largest* natural number that *divides* both $a$ and $b$, is called their *greatest common divisor*; we'll denote this by $a \vee b$.
As operations, the LCM and GCD have lots of nice properties. But two are important for us. For all natural numbers $a$, $b$, and $c$, we have the following.
1. $a \wedge b = \frac{ab}{a \vee b}$
2. $a \wedge (b \wedge c) = (a \wedge b) \wedge c$
The first property says that the LCM can be computed easily if we know the GCD. This is useful because there is a nice algorithm, called the Euclidean Algorithm, for computing GCDs.
Let $a$ and $b$ be natural numbers, and let $q$ and $r$ be natural numbers such that $a = qb+r$ and $0 \leq r < b$. Then $a \vee b = b \vee r$.