A number of people each wrote down one of the first 30 positive integers. Were any of the integers written down by more than one of the people?
(1) The number of people who wrote down an integer was greater than 40.
(2) The number of people who wrote down an integer was less than 70.
OA A
Source: Official Guide
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A number of people each wrote down one of the first 30 posit
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BTGmoderatorDC
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fskilnik@GMATH
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(simplified wording) FOCUS: Was at least one number "chosen" more than once (among the 30 numbers available)?BTGmoderatorDC wrote:A number of people each wrote down one of the first 30 positive integers. Were any of the integers written down by more than one of the people?
(1) The number of people who wrote down an integer was greater than 40.
(2) The number of people who wrote down an integer was less than 70.
Source: Official Guide
(1) Sufficient: if each person (among P people) would choose a different number, it would be needed at least P numbers.
We have 30 numbers to be chosen among P > 40 people. There are at least two people who will choose the same number. (*)
(2) Insufficient: we could have only 2 people, say A and B.
> If A chose 1 and B chose 2, the answer would be NO.
> If A chose 1 and B also chose 1, the answer would be YES.
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
(*) The Pigeon Principle : https://en.wikipedia.org/wiki/Pigeonhole_principle
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Jay@ManhattanReview
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To be certain that an integer, among the first 30, is repeated, at least 31 people must participate.BTGmoderatorDC wrote:A number of people each wrote down one of the first 30 positive integers. Were any of the integers written down by more than one of the people?
(1) The number of people who wrote down an integer was greater than 40.
(2) The number of people who wrote down an integer was less than 70.
OA A
Source: Official Guide
Let's take each statement one by one.
(1) The number of people who wrote down an integer was greater than 40.
Since in Statement 1, we have 40 people ≥ 31 people, it's certain that at least one integer is written down two people. Sufficient.
(2) The number of people who wrote down an integer was less than 70.
Case 1: If there are less than 31 people and each wrote a different integer, the answer is No.
Case 2: If there are greater than 31 but less than 70 people, the situation in Statement 1 is applicable; the answer is Yes.
No unique answer. Insufficient.
The correct answer: A
Hope this helps!
-Jay
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Brent@GMATPrepNow
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Target question: Were any of the integers written down by more than one of the people?BTGmoderatorDC wrote:A number of people each wrote down one of the first 30 positive integers. Were any of the integers written down by more than one of the people?
(1) The number of people who wrote down an integer was greater than 40.
(2) The number of people who wrote down an integer was less than 70.
Given: A number of people each wrote down one of the first 30 positive integers.
Statement 1: The number of people who wrote down an integer was greater than 40.
If there were exactly 30 people, then each person could write a different number (i.e., the 1st person writes 1, the 2nd person writes 2, the 3rd person writes 3, . . . the 30th person writes 30)
However, when the 31st person goes to write a number, he/she will inevitably write a number that has already been written.
So, once there are more than 30 people, we know there will be at least one duplicate number.
So, if there are 40 people, then the answer to the target question is YES, at least one of integers was written down by more than one person
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: The number of people who wrote down an integer was less than 70.
There are several possible scenarios that satisfy statement 2. Here are two:
Case a: There are 40 people (i.e., statement 1). In this case, the answer to the target question is YES, at least one of integers was written down by more than one person
Case b: There are only 2 people and they write 3 and 5 respectively. In this case, the answer to the target question is NO, no integer was written down by more than one person
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
