GMAT Club World Cup 2022 (DAY 2): Is the range of a set of hundred

GMAT CLUB Official Explanation:

Is the range of a set of hundred distinct even integers less than 200?

First of all notice that we are told that the numbers in the set are distinct and even.

Next, lets' check what is the minimum possible range of a set with 100 distinct even integers. Well, the minimum range would be of the integers are consecutive, in this case the set will be least spread. The range of any set of 100 consecutive even integers is {Range} = {Largest} - {Smallest} = (x + 198) - x = 198. Any other set of 100 distinct even integers will have the range of 200 or more. So, the question basically asks whether the set consists of even consecutive integers.

(1) The product of all the numbers in the set is negative.

All of the numbers in the set cannot be negative, because in this case the product of 100 (even) negative integers will be positive. So, the set must have at least one positive number (and odd number of negative numbers). Also, the set cannot have 0 in it, because in this case the product will be 0, not negative.

So, we get that the set has both negative and positive numbers in it but not 0. Thus, the numbers in the set cannot be consecutive even integers (we cannot skip 0, which is even). As discussed above, if the numbers in the set are not even consecutive integers, then the set has the range of 200 or more. Sufficient.

(2) The product of the smallest and the largest numbers in the set is negative.

This means that the smallest number in the set is negative and the largest number in the set is positive. The numbers in the set could be consecutive integers, for example, {-2, 0, 2, 4, ..., 196} but they also could not be consecutive integers, for example, {-2, 100, 102, 104, ..., 296}. Not sufficient.

Answer: A.