For integers 𝑎, 𝑏 and 𝑐, what would be the minimum and maximum values respectively of 𝑎 + 𝑏 + 𝑐 if log |𝑎| + log |𝑏| + log |𝑐| = 0?

The correct option is -3 and 3.

To find the minimum and maximum values of the sum 𝑎 + 𝑏 + 𝑐, let us analyze the given equation:
$$\log \left|a\right|+\log \left|b\right|+\log \left|c\right|=0$$

Using the logarithmic property log(x) + log(y) + log(z) = log(x · y · z), we can combine the terms:
$$\log \left(\right|a|\cdot |b|\cdot |c\left|\right)=0$$

By taking the exponential of both sides (with respect to the base of the logarithm, typically 10 or e), we get:
$$\left|a\right|\cdot \left|b\right|\cdot \left|c\right|=1$$

Since 𝑎, 𝑏, and 𝑐 are restricted to be integers, their absolute values |𝑎|, |𝑏|, and |𝑐| must also be integers. The only way the product of three positive integers is equal to 1 is if each individual integer has an absolute value of 1.
Therefore, we must have:
$$\left|a\right|=1,\left|b\right|=1,\left|c\right|=1$$

This implies that each of the variables 𝑎, 𝑏, and 𝑐 can independently take the values of either -1 or 1:
$$a\in \{-1,1\},b\in \{-1,1\},c\in \{-1,1\}$$

Now, we find the range of values for the sum 𝑎 + 𝑏 + 𝑐:
1. Minimum Value: The sum is minimized when all three variables are chosen to be their smallest possible value, which is -1.
$$\text{Minimum Sum}=(-1)+(-1)+(-1)=-3$$
2. Maximum Value: The sum is maximized when all three variables are chosen to be their largest possible value, which is 1.
$$\text{Maximum Sum}=1+1+1=3$$

Thus, the minimum and maximum values of 𝑎 + 𝑏 + 𝑐 are -3 and 3, respectively.