A pirate captain has 63 bags of coin with the same amount of coin inside each of the bags. If he wants to divided the coins to his 23 henchman evenly, he has to add 7 more coins. How many coins inside each bags?
P.S. I have this question after learning about diophantine equation, so maybe I have to use that. But I don't know where to start.
Let each of the bags have x coins. Now, the idea is that if he is dividing it among 23 people, 23 is a divisor of the total number of coins, since one can safely assume that he is not going to split a coin. So, you need to look at how many coins there are in total, once he has added 7 coins. That is exactly 63*x + 7. Now, you need an x for which 23 divides 63*x + 7. Therefore, it is clear that @lulu is correct in saying that there are multiple solutions and in fact, there are infinitely many (since if x is a solution, so is 23n + x, for any natural number n).
The problem is to solve: For what x is 63*x congruent to -7 (or 16), modulo 23. This is the same as asking for what x is 17*x congruent to 16, modulo 23. Solve for x.