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

ans (A) statement 1 alone is sufficient.

1. indicate that either one element is positive or one element is negative.
case 1 one element is negative
( -400,2,4,6,8............198)
raNGE >> 200
(-2,2,4,6........198)
range =200
sufficient

case 2 one element is positive
(-198,-196..............2)
range = 200

(-400,............2)
range>> 200
sufficient

2. indicate largest element is positive smallest is negative
( -2,..............198)
range = 200

( - 50............... 50)
range 100

not sufficient

The key to solving this is to understand that '0' is an even integer, and that since integers are mentioned, we can use negative numbers too.

Also, we know that Range = Highest - Lowest

Using (A), we can be sure to eliminate 0 for the set. When that happens, even if you take consecutive even integers, the range will be at least 200.

Using (B), we just know that the smallest number is negative and the largest number is positive. This doesn't really tell us anything since, if we take a set of 200 distinct even integers, say, (2, 0, -2, -4, -6,..., -196), the range is 198. But the lowest number can be -212, for example. In that case, the range is 214. So, the range can be less than 200, 200, or more than 200. We can't say for sure.

Whereas with option A, we can certainly say that the range shall be 200 or more but never less.

Hence, Option A is the right answer.

The correct option is A.

Statement 1 tells me the the even distinct integers do not include number 0. Therefore the range is not less than 200.
The even numbers are distinct and include at least 1 negative even integer and at least 1 positive integer, therefore the range in this case is at least 200.

Statement 2 does not provide enough information for me to answer the question.
The range can be less than 200, 200 or more than 200.

S1: "The product of all the numbers in the set is negative."

If product of all numbers is negative, number of negative numbers must be odd and none of the numbers is 0. And for each of the 100 numbers to be distinct, and even the minimum range possible is 200.

E.g.
{-2,2,4,6,8, .. 198} in ascending order => Range = 198-(-2)=200
{-6,-4,-2,2,4,6... 194} in ascending order => Range = 194-(-6)=200
If we increase the absolute value of the maximum and minimum, the range will only increase
{-100,-2,-4,2,4,6,...194} in ascending order => Range=194-(-100)=294 > 200
This we see range can never be less than 0 in this case. So statement is SUFFICIENT.

S2: "The product of the smallest and the largest numbers in the set is negative."

In this case one of the number (not the smallest or largest) can be zero. In that case there is a possibility that the range may be less that 200
E.g.
{-2,0,2,4,6,8..196} => Range = 196-(-2)=198 < 200
another case,
{-2,2,4,6,8, .. 198} in ascending order => Range = 198-(-2)=200
So statement is INSUFFICIENT.

Hence answer is A

1)Now the product to be negative the number of negative integers in the set should be odd and it should not contain 0
So if the set starts with [-2,2,...198] - the range is 200
Again if we start the set with [-400,-398,-396,..,-2,2] -- the value would be greater than 200
Thus the range is always >=200.
B,C,E can be eliminated
2) This can include 0 as well as
So for a set with [-2,0,..196] - the range is less than 200
And for a set with out 0 [-2,2,--198] the range is equal to 200
Thus 2 is insufficient and D is out
Correct choice is A

Answer: A

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

If we consider the set the first hundred distinct even positive integers, the set will have numbers
(2, 4, 6, ...., 196, 198, 200)
Range for this set is 200 - 2 = 198
So the minimum range of a set of hundred distinct even integers will be 198.

(1) The product of all the numbers in the set is negative.
Lets consider -2 as one the number of the set and rest of the numbers be +ve multiples of 2 i.e. (2, 4, ...196, 198). This will make product as negative.
0 cannot be in the set as product needs to be negative.
This set will have the smallest range. i.e. 198 -(-2) = 200
For any other similar set (negative product) the range will be more than 200.
Sufficient. range will not be less than 200.

(2) The product of the smallest and the largest numbers in the set is negative.
If we consider the same set as in option 1, it will have product of smallest and largest numbers as negative. ( -2*198 = -ve)
In this case, range = 200.
Consider another set, (-2, 0, 2, 4, ....194, 196). In this case as well product of smallest and largest number is negative. But range = 196 -(-2) = 198.
For any other set, the range can be more than 200.
Insufficient. as range can be more than, less than or equal to 200.

Statement 1 alone is sufficient.

Set contains hundred DISTINCT even integers and we need to determine if the range of this set is less than 200?

An even integer is any integer which is completely divisible by 2 with 0 remainder, and that includes positive, negative and non-negative(0) integers {Ex: -20, -2, 0, 2, 20, 200 etc.}

Taking any set of 100 even integers where range can be less than 200:
1. All positive: From 2 to 200
2. All negative: From -2 to -200
3. All non negative: From 0 to 198
4. All non positive: From 0 to -198
5. Mixed Set 1: -2,0, And then From 2 to 196
6. Mixed Set 2: -2, And then from 2 to 198
Now if you examine all cases above, you can see that the first 5 all scenarios where range is less than 200 is possible but not in the last case. Here is the significance of 0. In the last 2 options, one has 0 in it and another does not and the one which has 0 satisfies range less than 200 and the other does not
So we need to examine the statements based on the conditions above and see how things pan out


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

Ok, so product of all numbers in the set is negative and since there are an even number (100) of terms in the set, this implies the following
a) Odd number of negative even integers
b) Set contains both positive and negative even integers
c) Set does not contain 0 because else the product would be equal to 0
So based on these, we can safely assume that this falls under category 6 (6. Mixed Set 2: -2, And then from 2 to 198
) mentioned above in which case range cannot be less than 200

SUFFICIENT

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

Ok, so product of smallest and largest numbers is negative which implies the following:
a) Both positive and negative even integers
b) Can or cannot contain 0

So basically this falls under either category 5 or category 6 (5. Mixed Set 1: -2,0, And then From 2 to 196
6. Mixed Set 2: -2, And then from 2 to 198). So range can be less than 200 in case of 0 present in set and cannot be less than 200 if 0 is not present in the set

NOT SUFFICIENT

Answer - A

(1) The product of all the numbers in the set is negative. Therefore the set doesn't contain 0.
Example -2; 2; 4; ... 198 => The minimum range is 200 => The range of a set is not less than 200.
=> Suff

(2) The product of the smallest and the largest numbers in the set is negative.
=> The set can contain 0 or not.
If the set contains 100 consecutive integers from -2 to 196 => the range is 198, less than 200.
However, the set can have a wider range, for example from -10 to 200 => the range is larger than 200.
=> Insuff

=> Choice A

Imo A

Statement1: Exclude 0 in the set
For set { -2,4.... 198} the range is 200
For set {-2,4....240}, the range is 242
For both the sets the answer is No
Sufficient

Statement2: Include 0 in the set
For set { -2,0, 4.... 196} the range is 198
For set {-2, 0, 4....240} the range is 242
Insufficient

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

Let’s first clarify that the range of hundred DISTINCT EVEN numbers will be less than 200 only when they are consecutive even numbers, only in such case their range is 198. How to understand this:

from 1 to 100, there are 100 numbers. Multiply all by 2 and you will get hundred even consecutive numbers, such as 2, 4, 6, 8, … ,200. The range is 200 – 2 = 198 > 200 (the set may begin with any number, even negative number, but the set should be consecutive)

In any other case the range is NOT lower than 200.


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

Case 1: If all 100 even numbers are negative and consecutive, then their range is 198 and their product is POSITIVE.
Case 2: If all 100 even numbers are positive and consecutive, then their range is 198 and their product is POSITIVE.
Case 3: If all 100 even numbers are consecutive and include both negative and positive numbers, then their range is 198 and their product is ZERO.
Case 4: Since statement A says that their product is NEGATIVE, it’s clear that they are not consecutive, which mean that the range is not less than 200.
Sufficient

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

This statement may imply either Case 3 or Case 4 above. So, insufficient.

So A

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