To find the least positive sum of three integers whose product is 60, let's analyze possible factorizations of 60. We want three integers $a$, $b$, and $c$ such that:

$a \times b \times c = 60$

and the sum $a + b + c$ is minimized.

Step 1: Factorizations of 60

The integer 60 can be factored in different ways. Let’s examine combinations of three integers that multiply to 60.

Some possible factor combinations of 60 are:

1. $1 \times 5 \times 12$ with sum $1 + 5 + 12 = 18$
2. $1 \times 6 \times 10$ with sum $1 + 6 + 10 = 17$
3. $2 \times 3 \times 10$ with sum $2 + 3 + 10 = 15$
4. $2 \times 5 \times 6$ with sum $2 + 5 + 6 = 13$
5. $-3 \times -4 \times 5$ with sum $-3 + (-4) + 5 = -2$ (not positive)
6. $3 \times 4 \times 5$ with sum $3 + 4 + 5 = 12$

Step 2: Select the Combination with the Least Positive Sum

From the factorizations above, the combination $3 \times 4 \times 5$ yields the minimum positive sum of $3 + 4 + 5 = 12$.

Conclusion

The least positive sum of three integers whose product is 60 is:

$\boxed{12}$

Would you like further details on any of these steps, or do you have additional questions?

Here are five related questions for further exploration:

1. How would the answer change if the product were a different integer, such as 120?
2. What if we allowed negative integers and still aimed for the least absolute sum?
3. Can we find the least positive sum if the product is a negative integer?
4. How can this approach be generalized for the product of four integers?
5. Are there efficient algorithms to find integer sets for a given product and minimum sum?

Tip: When trying to minimize or maximize integer sums for a given product, always start by examining small integer factors and systematically work up.