3000 = 2^3 X 3 X 5^3 = 8 x 3 x 125 Say a = 8, be = 3, and c = 125 Since the LCM comes from the largest of those, you need to find ways to distribute those primes. Let's start by finding how those can be combined. We can start looking at combinations that work, but we might miss some unless we have a good, methodical approach.
Since the prime powers are 3, 1, and 3, the number of factors of 3000 is 4x2x4 = 32. You take the product of the (powers + 1). So you can find any factor by taking 1 number from each of the following rows: 1 2 4 8 1 5 25 125 1 3 Let's look at all the factors, and narrow them down to the 3 digit factors.
From the first two rows above, you get the 16 factors 1 2 4 8, 5 10 20 40, 25 50 100 200,125 250 500 1000 -- those are also the factors you get by choosing a 1 from the third row. The factors obtained by choosing 3 are 3 times those numbers, are these 16 more 3 6 12 24,15 30 60 120, 75 150 300 600, 375 750 1500 3000 There are 11 3-digit numbers are 100 200 125 250 500 120 150 300 600 375 750 The table below shows the LCM of every combination of those, with the 3000 values highlighted. You should verify that in every one of those, 8, 125, and 3 will be a factor of one number in the pair.
I used Excel to get these LCM's, since I know that will be accurate, so I wouldn't miss any. You could do these by hand, but it would be more tedious. Actually you don't have to calculate all of them.
You can rule out those where 3, 8 or 125 are missing as factors. E.g. , 200 and 125 has an 8 as part of 200, and 125 as part of 125, but there is no 3.
So it can be skipped. I think the key to a complete and accurate answer is making sure the method doesn't miss any, and that's why I did this in a way that might seem elaborate, and a bit of overkill. But it's easier to just check them all, IMHO.
Edit: Oops, I trimmed the array incorrectly last night. Here's a corrected chart. Think about how you would characterize the numbers that work.
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