|Spoiler Inside: Solution to Puzzle||SelectShow>|
One way to solve this is to add each condition one at a time. First, the problem is a number equivalent to 1 (mod 2). Then, such a number must also be equivalent to 2 (mod 3). The solution to that is 5 (mod 6). Then, such a number must also be equivalent to 3 (mod 5). The solution is 23 (mod 30). Continue in this manner until the solution of 29243 (mod 30030) is found. The next matching value after 29243 is 29243 + 30030 which is over 50000, so 29243 is a unique solution.