What's the smallest integer which is divisible by all integers $1, 2, \dots, n$? Is there a simple way to represent the answer? Call it $f(n)$ here.
Clearly factorial ($n!$) satisfies the condition of divisibility. But once you get to $f(4)$ (which is $12$), factorial ($24$) is too large, because it includes as factors both $4$ and $2$, which are redundant.
Multiplying all the prime numbers up to $n$ gives you another estimate, which is this time too low, because you need to include primes multiple times when they're repeated in the factorization of the inputs.
So the answer is somewhere between factorial, and the product of primes. Is there a simple answer?