What is the Positive Integer n? GMAT Data Sufficiency

Question: What is the positive integer n?

1. The sum of all of the positive factors of n that are less than n is equal to n
2. n < 30

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Correct Answer: E
Solution and Explanation:
Approach Solution 1:
S2: Insufficient: We can start with the second statement first because it is clear that it is insufficient to solve the question what is the value of the positive integer n?
(1) Insufficient: We must first understand what this statement is saying. If all of the n’s factors (other than n itself) are added up, they equal n
We can begin our search by considering prime factors. It is impossible that the factors “other than the number” add up to the number for any prime number. Thus we can begin our search for such n’s with the number 4
4 does not equal 1 + 2
6 does not equal 1 + 2 + 3
9 does not equal 1 + 3
10 does not equal 1 + 2 + 5
12 does not equal 1 + 2 + 3 + 4 + 6
14 does not equal 1 + 2 + 7
15 does not equal 1 + 3 + 5
At this point we might be tempted to think that this is a property that is unique to 6 and is unlikely to come around again. It would behoove us to keep searching though and to atleast cover the range defined by the second statement (i.ie n < 30). If we do that we see that this property repeats itself one other time in the remaining integers that are less than 30
16 does not equal 1 + 2 + 4 + 8
18 does not equal 1 + 2 + 9
20 does not equal 1 + 2 + 4 + 5 + 10
21 does not equal 1 + 3 + 7
22 does not equal 1 + 2 + 11
24 does not equal 1 + 2 + 3 + 4 + 6 + 8 + 12
25 does not equal 1 + 5
26 does not equal 1 + 2 + 13
27 does not equal 1 + 3 + 9
28 does not equal 1 + 2 + 4 + 7 + 14
Hence Statements (1) and (2) TOGETHER are NOT sufficient.

Approach Solution 2:
The problem statement asks to find the value of N
Let solve the problem by Test cases method
Statement 1: The sum of all of the positive factors of n that are less than n is equal to n
=> n can be 4 or 3 or anything else. Hence, Not sufficient.
Statement 2: n < 30
=> We cannot derive the value of n with certainty. Hence clearly insufficient
Combining statement 1 and statement 2 we get:
N can be 2 or 3 or 4 => Hence, both statements together are not sufficient.

Approach Solution 3:
The problem statement asks to find the value of N
Statement 1 alone: The sum of all positive factors of 'n' that are less than 'n' is equal to 'n'.
A perfect number is a positive integer that is equal to the sum of its positive factors.
For instance: 6, 28, 478, etc.
This statement implies that n = positive numbers = 6, 28, 478, … and so on.
However, the information provided is not sufficient to get at a definite value in the range of perfect numbers.
Hence, statement 1 alone is NOT SUFFICIENT.
Statement 2 alone: n < 30
This means that 0 < n < 30. 'n' can be any number within the range of positive integers between 1 and 30. For example: 1, 2, 3, ..., 29.
However, the information provided is not sufficient to get at a definite value of n.
Hence, statement 2 alone is NOT SUFFICIENT.
Combining both statements together:
Statement 1: n is a perfect number
Statement 2: n < 30
Therefore, 'n' is a perfect number that is less than 30 which means 'n' can either be equal to 6 or 28.
Since the exact value of 'n' remains unknown, then both statements combined together are NOT SUFFICIENT.

“What is the positive integer n?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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