SOLUTION: The positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?

Question 541373: The positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

30, 72, and N

There must exist positive integers A, B, and C such that 

30�72 = NA
  30N = 72B
  72N = 30C

Simplifying:

2160 =  NA
  5N = 12B
 12N =  5C  

Solving each for N:

N = 
N = 
N = 

Set the last two right sides equal, since both are equal to N

 = 

144B = 25C

The smallest B and C can be are B = 25 and C = 144

Substituting in

N =  =  = 60 
N =  =  = 60

That works because

30, 72, and 60

30*72 = 2160 = 60*36
30*60 = 1800 = 73*25
60*72 = 4320 = 30*144

So the answer is N = 60

Edwin

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