, 2020

Since $3 \times 5 \times 7 = 105$, then the greatest possible value of $n$ is at least 105.

In particular, the greatest possible value of $n$ must be positive.

For the product of three numbers to be positive, either all three numbers are positive (that is, none of the numbers is negative) or one number is positive and two numbers are negative. (If there were an odd number of negative factors, the product would be negative.)

If all three numbers are positive, the product is as large as possible when the three numbers are each as large as possible. In this case, the greatest possible value of $n$ is $3 \times 5 \times 7 = 105$.

If one number is positive and two numbers are negative, their product is as large as possible if the positive number is as large as possible (7) and the product of the two negative numbers is as large as possible.

The product of the two negative numbers will be as large as possible when the negative numbers are each “as negative as possible” (that is, as far from 0 as possible). In this case, these numbers are thus $-4$ and $-6$ with product $(-4) \times (-6) = 24$. (We can check the other possible products of two negative numbers and see that none is as large.)

So the greatest possible value of $n$ in this case is $7 \times (-4) \times (-6) = 7 \times 24 = 168$.

Combining the two cases, we see that the greatest possible value of $n$ is 168.