276: Primes' Product and Sum

The really difficult questions on the SAT (the ones at the end of each section) are often not really that difficult -- they just seem to be hard to answer because they're hard to read and understand because of the formal mathematical writing style and the complexity of the sentence. Often the best tactic is to plug in some numbers for the vague-sounding variables and make the problem more concrete.

"The positive number k is the product of four different positive prime numbers.
If the sum of these four prime numbers is a prime number greater than 20,
what is the least possible value for k?"

So what can we do with this one to make it simpler to understand the question?
Where would you start?