We are given the equation:

$A \times B \times C = 60$

where $A, B, C$ are integers (positive or negative), and we need to find the minimum possible positive value of $Y = A + B + C$.

Step 1: Find Factorizations of 60

We consider integer factorizations of 60:

$60 = 1 \times 2 \times 30$ $60 = 1 \times 3 \times 20$ $60 = 1 \times 4 \times 15$ $60 = 1 \times 5 \times 12$ $60 = 1 \times 6 \times 10$ $60 = 2 \times 3 \times 10$ $60 = 2 \times 5 \times 6$

Additionally, we can consider negative factors since their product should still give 60. For example:

$(-1) \times (-2) \times 30 = 60$

Step 2: Minimize $A + B + C$

We compute $A + B + C$ for various cases:

1. $(-1, -2, 30) \Rightarrow (-1) + (-2) + 30 = 27$
2. $(-1, -3, 20) \Rightarrow (-1) + (-3) + 20 = 16$
3. $(-1, -4, 15) \Rightarrow (-1) + (-4) + 15 = 10$
4. $(-1, -5, 12) \Rightarrow (-1) + (-5) + 12 = 6$
5. $(-1, -6, 10) \Rightarrow (-1) + (-6) + 10 = 3$ (smallest positive integer found so far)
6. $(-2, -3, 10) \Rightarrow (-2) + (-3) + 10 = 5$
7. $(-2, -5, 6) \Rightarrow (-2) + (-5) + 6 = -1$ (not positive)

Conclusion:

The smallest positive $Y$ is 3, which is achieved by the values:

$A = -1, B = -6, C = 10$

Thus, the minimum possible value of $Y$ is:

$\boxed{3}$

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Would you like me to explain any step further? Here are some related questions:

1. How do we systematically find integer factorizations of a number?
2. What if the constraint required $A, B, C$ to be strictly positive?
3. Can we generalize this method for any given product $P$?
4. How does changing the sign of a factor affect the sum?
5. How would we solve a similar problem with four variables instead of three?

Tip: When minimizing sums in integer factorization problems, consider negative factors to balance values toward the smallest positive sum.