Find a six-digit number whose first three are 637 and that is divisible by 21,23, and 24.
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If it's divisible by 21, 23, and 24, you can start by multiplying those together:
21 * 23 * 24 = 11592
Then you can multiply that by numbers until you're in the right realm. In this case, it's a number starting with 11, and you need it to start with 637. If you divide 637/11 you get 57.909 or so, so you have a starting point: let's multiply 11592 by 56.
That gives us 649152, which is slightly higher than we want it. If we multiply by 55, we get 637560. So:
21 * 23 * 24 * 55 = 637560
We know it's divisible by the three numbers, because they were multiplied in to produce the number, and this number starts with 637. Eureka!