Product of 3 integers a, b, c equals 72, where every factor is positive integer. Find the integers a, b, c with the smallest sum.
It's easy to get the factors of 72 manually and see that the 3 smallest factors that give the product as 72 will be the smallest sum.
I checked the factorization like this:
1-1-72: 74
1-2-36: 39
1-4-18: 25
1-8-9: 18
2-4-9: 15
3-4-6: 13
It seems like the smallest possible factors will give the smallest sum.
But there must be some trick to it or some kind of algorithm to find the smallest integers for any arbitrary product without laborious factoring.
Thanks for your help.
Best Answer
You could try decomposing $72$ into its prime factors. This is:
$$72 = 2^3\times 3^2 = 2*2*2*3*3.$$
Then pick the smallest combination of factor, which is $2\times 2$, $2\times 3$ and $3$, so that $4\times 6\times 3 = 72$, and $4 + 6 + 3 = 13$.
Hope it helps.