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What is the positive integer n?

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What is the positive integer n?  [#permalink]

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New post Updated on: 12 Jun 2013, 03:15
3
7
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

48% (01:38) correct 52% (01:46) wrong based on 367 sessions

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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

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Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730


Originally posted by enigma123 on 27 Jan 2012, 15:44.
Last edited by Bunuel on 12 Jun 2013, 03:15, edited 2 times in total.
Added the OA
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Re: What is n?  [#permalink]

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New post 27 Jan 2012, 16:00
6
5
enigma123 wrote:
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

For me its E. Can someone please help and let me know if its right or wrong?


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.
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Re: What is the positive integer n? (1) The sum of all of the  [#permalink]

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New post 27 Jan 2012, 16:05
Thanks Bunuel. I will leave this question then.
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Re: What is the positive integer n?  [#permalink]

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New post 24 Aug 2016, 03:32
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
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Re: What is the positive integer n?  [#permalink]

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New post 27 Dec 2016, 04:14
1
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.
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Re: What is the positive integer n?  [#permalink]

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New post 19 Mar 2018, 22:24
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Re: What is the positive integer n? &nbs [#permalink] 19 Mar 2018, 22:24
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