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Manhattan GMAT 700+ Challenge Problem

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by Kevin » Mon Jun 12, 2006 8:54 am
Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer (highlight to read)

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 was 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.
Kevin Fitzgerald
Director of Marketing and Student Relations
Manhattan GMAT
800-576-4626

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by harsh.champ » Fri Mar 12, 2010 7:40 am
Kevin wrote:Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer (highlight to read)

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 was 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.
I could not complete this question in 2 min.
I find that many manhattan challenge questions take more than 2 min. :(
It takes time and effort to explain, so if my comment helped you please press Thanks button :)



Just because something is hard doesn't mean you shouldn't try,it means you should just try harder.

"Keep Walking" - Johnny Walker :P
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by gmatmachoman » Fri Mar 12, 2010 9:33 am
@Manhattan,

I am of the opinion these may not really "pop-up" in GMAT.Reason would be GMAC tests some "fundamental logic/concept".

If we could "fabricate" tough questions then India's CAT(The toughest MBA admission test for IIM's) quant will also fit in to GMAT ..right??

But just by framing tougher questions' what does MGMAT really achieves?? What's the real strategy if these questions are seldom tested in real GMAT??

Can u plz shed some "light" in my query!

I would love to hear your version of reasoning for the fundamental doubt every one has that" Why MGMAT Quants are"framed" tougher than real GMAT?

Rgds
Govardhan.R
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by hooliganpete » Fri Mar 12, 2010 10:36 am
The poster says that the Answer is E but discovers that the number 28 satisfies both statement 1 and 2. Didn't you mean C as the correct answer?[/quote]
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by gmatmachoman » Fri Mar 12, 2010 12:57 pm
hooliganpete wrote:The poster says that the Answer is E but discovers that the number 28 satisfies both statement 1 and 2. Didn't you mean C as the correct answer?
[/quote]

No mate,OP asks for only one value for n satisfying the stated condition. But we can find more than one value using st1.So st1 not sufficient\\

For st2,again we have 2 distinct values of "n" (6&28) so we cannot precisely say the value of n.
Insufficient

SO E

But IMO,these cant come in GMAT,as I have seen GMAC tests some mathematical logic..here i see only addition...
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by diebeatsthegmat » Wed May 12, 2010 12:04 pm
Kevin wrote:Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer (highlight to read)

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 was 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.
oh, man. thank a lot for this explaination. my answer is also E but was not so sure. anyways your explanation helped my stagnant brain a lot. thanks...
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by vikas.yaadav » Mon Jun 21, 2010 1:14 pm
diebeatsthegmat wrote:
Kevin wrote:Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer (highlight to read)

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 was 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.
oh, man. thank a lot for this explaination. my answer is also E but was not so sure. anyways your explanation helped my stagnant brain a lot. thanks...
This certainly falls into the category of question's where you guess and move on, unless you know the answer from start.
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by nikhilkatira » Tue Jun 22, 2010 5:00 am
machoman bhai...i guess these numbers such as 6 ,28... have certain names...and they fall into some concept category and just addition.

My question is does GMAT tests such concepts ?
Best,
Nikhil H. Katira
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by sreak1089 » Fri Jun 25, 2010 12:49 am
Perfect numbers. Numbers such as 6, 28, 496, 8128, etc are called perfect numbers because, their factors (excluding the original number) add up to the original number.
nikhilkatira wrote:machoman bhai...i guess these numbers such as 6 ,28... have certain names...and they fall into some concept category and just addition.

My question is does GMAT tests such concepts ?
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by nikhilkatira » Fri Jun 25, 2010 1:39 am
sreak1089 wrote:Perfect numbers. Numbers such as 6, 28, 496, 8128, etc are called perfect numbers because, their factors (excluding the original number) add up to the original number.
nikhilkatira wrote:machoman bhai...i guess these numbers such as 6 ,28... have certain names...and they fall into some concept category and just addition.

My question is does GMAT tests such concepts ?
yup....thnx sreak1089
Best,
Nikhil H. Katira
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