M17-27

Official Solution:


Is the product of four consecutive even integers positive?

Firstly, the product of four consecutive even integers can be either 0 (when one of the even integers is zero) or positive (when all integers are positive or when all integers are negative).

(1) The sum of these integers is positive but smaller than 20:

If the smallest term is -4 or less, for example {-4, -2, 0, 2}, then the sum will not be positive. Hence, the smallest term must be greater than -4.

If the greatest term is 8 or more, for example {2, 4, 6, 8}, then the sum will be 20 or greater. Therefore, the greatest term must be less than 8.

Based on this statement, there are only TWO possible sets: {-2, 0, 2, 4} and {0, 2, 4, 6}. The product of the terms in either of these two sets is zero, so the answer to the question is NO. Sufficient.

(2) The product of the middle two of these integers is positive.

All four integers can be positive, for example, consider {2, 4, 6, 8}, giving a YES answer (product = positive). However, we can also have the set where the smallest integer is 0: {0, 2, 4, 6}, giving a NO answer (product = 0). . Not sufficient.


Answer: A