I am a little confused on the topic and have a few questions to get me started with a larger assignment. Thanks!

1) What kinds of numbers have exactly 3 divisors? 4 divisors? and so on...

2) Do bigger numbers necessarily have more divisors?

3) Is there a way to figure out how many divisors 1,000,000 has without listing and counting them? The same for 1,000,000,000?

4) What's the smallest number that has 20 divisors?

No, prime numbers can only be divided by themselves and 1.