The product of 5 different temperatures is 12. If all of them are integers,…

To build a set of integers with a small target product using many factors, use two things together: (1) the sign rule -- a product of integers is positive when the number of negative factors is even; (2) padding factors of +-1 contribute only a sign, not extra magnitude, so they let you increase the count of integers used without changing the product's value.

1. List each option's integers and count them: the problem requires exactly five pairwise-distinct integers.

2. Check the count first: one option lists only four values, so it is eliminated immediately regardless of its product.

3. Multiply the first valid option's five integers in order: (-2) x (-1) = 2; 2 x 1 = 2; 2 x 2 = 4; 4 x 3 = 12. This matches the required product of 12.

4. Multiply the other five-integer option in order: (-4) x (-1) = 4; 4 x 1 = 4; 4 x 4 = 16; 16 x 9 = 144. This does not match 12.

5. Since a valid five-integer set exists, the 'no such set exists' option is also eliminated.

Independently verify the sign: the valid set has exactly two negative integers; an even count of negative factors always yields a positive product, consistent with the required positive value of 12. All five integers in that set are also pairwise distinct, satisfying the 'different' condition. As a separate magnitude check, taking absolute values of those five integers gives 2, 1, 1, 2, 3, whose product is 2 x 1 x 1 x 2 x 3 = 12 -- confirming the magnitude independently of the sign check.

Therefore, the set -2, -1, 1, 2, 3 is the unique valid answer among the given choices.