Finding smallest and largest prime factor of $\frac{200!}{180!}$
Note $\dfrac{200!}{180!}=200(199)\cdots(181)$. The smallest prime factor is easy to calculate - it's just $2$, since $2$ is the smallest prime and the product contains even factors. The largest prime factor is a little more interesting. You have to find the largest prime factor out of any of the elements in the product. However, noting that $199$ is prime, $200$ has no prime factors greater than $199$, and $199$ is greater than all other elements in the product - never mind their prime factors - gives us the result that the largest prime factor is $199$.