Question 499191: The sum of four natural numbers a1, a2, a3 and a4 is
210. What is the minimum possible LCM of these 4 mentioned
natural numbers?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Note that 210 = 2*3*5*7. Without loss of generality assume that  . The optimal case would be "a1, a2, a3 are factors of a4" in which the LCM would be simply a4. If this is true, then we can construct casework in which we find the minimum possible value of a4.
If a4 is strictly greater than a3, then a3 is at most 1/2 of a4, and
 . The optimal case would be 84, 42, 42, 42.
The next case would be "a4 = a3" and a2 is strictly less. Then,
Or, if a4 = a3 = a2, then
We can check that {30, 60, 60, 60} satisfies, and their LCM is 60.
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