For 5.19, I'm not sure about where to start: I don't know how to use Bezout's Identity
For 5.20: I'm guessing that I should factor it out? But how could I find its factorization?
For 5.21: 
First prove that there are at least 2 2's, we do this by trying out evens and odds-- if all are even, all of there differences are even as well; one odd, the three evens have three even differences; 2 odd, evens and odds have even differences respectively; 3 odd, three even differences; all odd, all even differences. Therefore, there exists at least 2^2 in the polynomial.
Then we prove that there is at least one factor of three: first, to not have a difference divisible by 3, every difference must be different mod 3. If there is for example, b-a and c-a =1 mod 3, then b-c=0 mod 3. Therefore, all six differences must be different mod 3 for there to not exist a factor of 3. However, a number can only = 0, 1, 2 mod3. Therefore, there is at least one factor of 3
Therefore 12 divides the polynomial

For 5.22: I don't know where to start
For 5.23:  I don't know where to start
For 5.24: 13 is not a very random number, the divisibility rule tells us that a number is divisible by 13 if the last three digits minus the first three digits is divisible by 13. I know how to count the total amount of numbers, however I do not know how to proceed.

Any hints would be appreciated!