Suppose the number 1155 is written as the product of three positive integers. what is the smallest possible value of the sum of those three integers?

My first thought when I read this problem was "11 is a factor of 1155."  Indeed,
1155 equals 11 times 105.  Next, determine factors of 105.  I see that 5 is a divisor of both 10 and 5, so that 105 equals 5 times 21.

Adding together these three integers:  11+5+21 = 37

But wait!  21 can be factored (e. g., into 7 times 3).

You'll need to identify all of the several sets of 3 positive integers each and find the sum of each set.  Only after you've done this can you choose the set whose sum of the three integers is the smallest.  Please show your work, so that others can give you feedback on it.

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