GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Nov 2019, 18:22 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # An infinite sequence of positive integers is called a perfect sequence

Author Message
TAGS:

### Hide Tags

Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 439
Location: United Kingdom
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

9
27 00:00

Difficulty:   95% (hard)

Question Stats: 37% (02:03) correct 63% (02:05) wrong based on 684 sessions

### HideShow timer Statistics

An infinite sequence of positive integers is called a perfect sequence. If each term in the sequence is a perfect number, that is, if each term can be expressed as the sum of its divisors, excluding itself. For example, 6 is a perfect number, as its divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.

_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Math Expert V
Joined: 02 Sep 2009
Posts: 59182
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

8
13
An infinite sequence of positive integers is called a perfect sequence. If each term in the sequence is a perfect number, that is, if each term can be expressed as the sum of its divisors, excluding itself. For example, 6 is a perfect number, as its divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a perfect sequence?

(1) Exactly one term in S is a prime number --> primes have exactly two divisors 1 and itself, hence no prime is a perfect number, which means that S is not a perfect sequence. Sufficient.

(2) In sequence S, each term after the first in S has exactly 3 divisors --> a number to have exactly 3 divisors must be square of a prime, for example 3^2=9 has 3 divisors: 1, 3, and 9 (1, p, and p^2). No, such number is a perfect number: 1+3 cannot equal to 9, (1+p cannot equal to p^2 for integer p), which means that S is not a perfect sequence. Sufficient.

Question about a perfect number: what-is-the-positive-integer-n-1-the-sum-of-all-of-the-126635.html

Hope it helps.
_________________
##### General Discussion
SVP  P
Joined: 24 Jul 2011
Posts: 1897
GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

2
(D) it is.

Using statement (1), if one term in the sequence is a prime number, that number can never have the sum of the divisors (except the number itself) add up to the number itself. Therefore that number is not a perfect number. Therefore the sequence containing this number is not a perfect sequence. Sufficient.

Using statement (2), if each term after the first has exactly three divisors, none of the numbers with these three divisors can be a perfect number. This is because one of the divisors will be the number itself, which will get excluded. The other two divisors will be 1 and another factor. This means the number must be = the factor + 1, which is not possible. Therefore this sequence is not a perfect sequence. Sufficient.

_________________

Awesome Work | Honest Advise | Outstanding Results

Reach Out, Lets chat!
Email: info at gyanone dot com | +91 98998 31738 | Skype: gyanone.services
Senior Manager  Joined: 24 Mar 2010
Posts: 335
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

1
An infinite sequence of positive integers is called a perfect
sequence if each term in the sequence is a perfect number,
that is, if each term can be expressed as the sum of its divisors,
excluding itself. For example, 6 is a perfect number, as its
divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a
perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.

The OA is D. I put down A as my answer. For statement 2 I get that a positive integer with 3 divisors will be the square of a prime number (4, 9, 25, 49, etc.). This statement though says that each term AFTER THE FIRST has 3 divisors. So the first term could be a perfect number or not. Maybe I'm missing something here. Any help on this will be helpful.
Kaplan GMAT Instructor Joined: 25 Aug 2009
Posts: 629
Location: Cambridge, MA
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

Arbitrageur wrote:
An infinite sequence of positive integers is called a perfect
sequence if each term in the sequence is a perfect number,
that is, if each term can be expressed as the sum of its divisors,
excluding itself. For example, 6 is a perfect number, as its
divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a
perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.

The OA is D. I put down A as my answer. For statement 2 I get that a positive integer with 3 divisors will be the square of a prime number (4, 9, 25, 49, etc.). This statement though says that each term AFTER THE FIRST has 3 divisors. So the first term could be a perfect number or not. Maybe I'm missing something here. Any help on this will be helpful.
Hi Arbitraguer,

The question asks if S is a perfect sequence, meaning that every term must be perfect. (1) tells us the first term is not perfect; (2) tells us that no term after the first can be perfect. Therefore, each of (1) and (2) answers "NO" to the question of whether the entire sequence is perfect. Both are sufficient!
_________________
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9795
Location: Pune, India
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

1
Arbitrageur wrote:
An infinite sequence of positive integers is called a perfect
sequence if each term in the sequence is a perfect number,
that is, if each term can be expressed as the sum of its divisors,
excluding itself. For example, 6 is a perfect number, as its
divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a
perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.

The OA is D. I put down A as my answer. For statement 2 I get that a positive integer with 3 divisors will be the square of a prime number (4, 9, 25, 49, etc.). This statement though says that each term AFTER THE FIRST has 3 divisors. So the first term could be a perfect number or not. Maybe I'm missing something here. Any help on this will be helpful.

In the sequence S, if there is even one term which is not a perfect number, the sequence is not a perfect sequence. You need every term of the sequence to be a perfect number for the sequence to be a perfect sequence.
Statement 2 tells you that after the first term, every term is 'non-perfect'. We don't care whether the first term is perfect or not. Since we know that the sequence has non-perfect numbers, the sequence is not perfect. Hence, statement 2 is also sufficient.

Test makers like to add little twists like these "after the first term" to mess with your mind! I am sure you would have had no problems if the second statement were "...each term has exactly 3 divisors"
_________________
Karishma
Veritas Prep GMAT Instructor

Math Expert V
Joined: 02 Sep 2009
Posts: 59182
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE
_________________
Manager  Joined: 14 Nov 2011
Posts: 114
Location: United States
Concentration: General Management, Entrepreneurship
GPA: 3.61
WE: Consulting (Manufacturing)
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

enigma123 wrote:
An infinite sequence of positive integers is called a perfect sequence. If each term in the sequence is a perfect number, that is, if each term can be expressed as the sum of its divisors, excluding itself. For example, 6 is a perfect number, as its divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors.

question can be written in a better way:

An infinite sequence of positive integers is called a perfect sequence, if each term in the sequence is a perfect number, that is, if each term can be expressed as the sum of its divisors, excluding itself. For example, 6 is a perfect number, as its divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a perfect sequence?
Manager  Joined: 28 Feb 2012
Posts: 103
GPA: 3.9
WE: Marketing (Other)
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

1
An infinite sequence of positive integers is called a perfect sequence. If each term in the sequence is a perfect number, that is, if each term can be expressed as the sum of its divisors, excluding itself. For example, 6 is a perfect number, as its divisors, 1, 2, and 3, sum to 6. Is the infinite sequence S a perfect sequence?

(1) Exactly one term in S is a prime number.
(2) In sequence S, each term after the first in S has exactly 3 divisors

1 st.) This statement sufficient by itself, because any the feature of the prime number is that it has only two divisors, 1 and the number itself. But according to the definition of the perfect numbers the sum of the divisors (excl. the number itself) should be equal to the number itself - which is not possible with prime numbers. So the sequence is not perfect. Sufficient.

2 st.) lets take some numbers that have exactly 3 divisors: 4 (1, 2, 4) - the sum of the 1+2 is 3, which is not perfect number. next number is 9 (1, 3, 9) the sum of 1+3=4 again not perfect since it does not equal to 9. Next number is 25 (1, 5, 25) the same conclusion. Here is the pattern, only squares of the prime numbers could have exactly 3 divisors, that means in this sequence we have not perfect numbers - sufficient.

So each statement sufficient on its own - D.
_________________
If you found my post useful and/or interesting - you are welcome to give kudos!
Intern  Joined: 05 Jun 2012
Posts: 46
Schools: IIMA
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

Indeed a very nice question !!!

Thanks again Bunnel for presenting solution with such a simplicity!!!
_________________
If you are not over prepared then you are under prepared !!!
Non-Human User Joined: 09 Sep 2013
Posts: 13620
Re: An infinite sequence of positive integers is called a perfect sequence  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: An infinite sequence of positive integers is called a perfect sequence   [#permalink] 15 Dec 2018, 05:55
Display posts from previous: Sort by

# An infinite sequence of positive integers is called a perfect sequence  