drill A) X/Y = i (i is integer} + 13/Y
Explanation: since 13 is a prime number and it cannot be divided by any integer except for 13 itself and 1, Y cannot be 1 and 13. In its turn X = i*Y + 13, where Y is again any number (integer) except for 13 and 1. There is no unique solution for this drill, as X can be 27 (1*14+13), 43 (2*15+13) ...
drill B)X/3 = i (i is integer} and X/8 = i (i is integer}, find Sigma (CM (common multiples) X, Y from 1-1000)-?
Explanation: According to general divisibility rules, the number can be divided by 3 only if it's divisible by 3, and the number is divisible by 8 only if it's divisible by 2 and 4 simultaneously; CM (common multiples) for 3 and 8 should be ones which could be factored to 2^3 and 3 OR 24 --> from 1 to 1000 there are 41 sets of 24 (2^3*4). Now if we find the possible ways of combination for 41 sets of 24, we get the result of (41-1)* 4C1 (this part combines 2^3*3, that is 4 numbers to be distributed by 1), 40*4=160. Although we are able to find the number of CM (common multiples) for X and Y, CM-s themselves could take more time to be found out... we need 41 sets of 24s and additional 160 sets of possible combination for common multiples of X and Y
1000/24 leaves remainder, hence the combination set above by using 41 sets of 24 needs to be corrected slightly

seuss88 wrote:Examples include:

If X/Y has a remainder of 13, what could be a value of X. Also, if X is divisible by 3, and Y is divisible by 8, what is the sum of all the common multiples [ of X and Y - I guess this part was implied too ] between 1-1000. These are each made-up problems, but these are examples of division problems that I find relevant. Thank you!

Thank you, that is very helpful. Aside from this site, which offers a handful of remainder problems, do you know any other resource I could use to ONLY do these problems? I am not only looking for problem solving, but also for data sufficiency. Thanks again!

gmat-club might be another great web-site to complement the BTG; one difference here, is that with the BTG we have many experts contributing to member posts with their competent solution/approach. I would seek practice with 4gmat web-site too. The latter is very helpful and offers original (meaning, not ordinary) GMAT-like problems and DS questions.
/Thanks

seuss88 wrote:Thank you, that is very helpful. Aside from this site, which offers a handful of remainder problems, do you know any other resource I could use to ONLY do these problems? I am not only looking for problem solving, but also for data sufficiency. Thanks again!