M38-03

Official Solution:


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, let's check what the minimum possible range of a set with 100 distinct even integers would be. The minimum range would occur if the integers are consecutive, as this would make the set the least spread out. 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 a range of 200 or more. So, the question essentially asks whether the set consists of consecutive even integers. Meaning that only if the set consists of 100 consecutive even integers, its range will be less than 200, and we'll have a YES answer to the question. However, if the 100 distinct even integers in the set are NOT consecutive, its range will be 200 or more, and we'll have a NO answer to the question.

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

So, we deduce 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 consecutive even integers, then the set has a range of 200 or more, and we have a definite NO answer to the question. 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 even integers, for example, {-2, 0, 2, 4, ..., 196}, in which case the range would be less than 200, but they also could not be consecutive even integers, for example, {-2, 100, 102, 104, ..., 296}, in which case the range would be 200 or more. Since both possibilities exist, we cannot determine a definitive answer to the question from this statement alone. Not sufficient.


Answer: A