What is the positive integer n?

Question

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

Bunuel · Accepted Answer

It's quite time consuming question for those who are not familiar with perfect numbers: perfect number is a number which equals to the sum of its proper divisors (a positive divisor, which is different from the number itself).

There are 2 small perfect numbers: 6=1+2+3 and 28=1+2+4+7+14, (the next two are 496 and 8,128). So as you can see both 6 and 28 satisfy statement (1), as well as (2), hence the answer is E, for sure. Though if you don't KNOW this fact then you should manually break integers less than 30 into factors (less than this integer itself) and check whether their sum equals to it, which is quite tedious work. Of course some logic and common sense can simplify this work, for example you can exclude primes right away, but still I don't think that you'll see anything like this on the real test.

enigma123 · Answer

Thanks Bunuel. I will leave this question then.

stonecold · Answer

Awesome Question  enigma123 
Here We need the value of N
I did it by Test cases method
In Statement 1=> n can be 4 or 3 or anything else => Not suff
Statement 2 => Clearly insuff
Combining them N can be 2 or 3 or 4 => Not suff

SMASH THAT E

anairamitch1804 · Answer

For those who are looking for multiple cases that can be formed. Please refer below solution.

There is no conceptual or formulaic approach for solving this question. One must simply try out various integers.

(2) INSUFFICIENT: We can start with the second statement first because it is clear that it is insufficient to solve the question what is value of the positive integer n?

(1) INSUFFICIENT: We must first understand what this statement is saying. If all of n's factors (other than n itself) are added up, they equal n.

We can begin our search by considering prime factors. By definition prime factors have only two factors, themselves and 1. 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 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 (i.e. that the answer is A). It would behoove us to keep searching though and to at least cover the range defined by the second statement (i.e. 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 EQUAL 1 + 2 + 4 + 7 + 14

The correct answer is E.

LevanKhukhunashvili · Answer

I just remembered that 6 is a perfect number but I assumed that the following one would be beyond 30 Trying to test nearly all numbers less than 30 is "sepoku" on real GMAT I marked C

pranav13 · Answer

@ANIARAMITCH1804

Don't you think the answer would be D since 6 is the only possible number which satisfies the given condition.

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What is the positive integer n?

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