I had a similar question with a divisibility/remainder problem that I came across today from the MGMAT Number Properties
X is a positive integer. If X is divided by 11, the quotient is Y and the remainder is 3. If X is divided by 19, the remainder is 3. What is the remainder when Y is divided by 19?
The way to solve this problem is to write the two equations (X=11Y+3 and X=19Z+3) and set them equal to each other so (11Y+3=19Z+3 --> 11Y=19Z).
Then you know that 11Y is divisible by 19 because z is an integer, and specifically, Y is divisible by 19 since 11 is not. Thus, the remainder is 0.
Is it correct to say that the way to solve divisibility/remainder problems is to write 2 equations, set them equal to each other, and then make divisibility deductions?