puzzle

[sandeep]

Product of three integers a, b, c equals 72, where every factor is positive integer. Find the integers a,b,c with the smallest sum.
With hit and trial, I can see the combination 2,6,6 or 3,3,8 gives the sum as 14.
Any other approach one can think of??

 

[Tsotne]

4,3,6

 

[pruse]

By AM-GM, the sum must be at least \(3\sqrt[3]{72}=12.48...\). But since they're integers, the sum must actually be at least 13. And as Tsotne shows, 13 is attainable.

 

[D C]

Look at the prime factorization of 72:
\(72 = 2^3*3^2\)

then choose the combination that sums to 13...3, 4, 6 is really \(3,2^2,2*3\)