Last visit was: 15 Jul 2024, 07:44 It is currently 15 Jul 2024, 07:44
Toolkit
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

SORT BY:
Tags:
Show Tags
Hide Tags
Senior Manager
Joined: 25 Jun 2011
Status:Finally Done. Admitted in Kellogg for 2015 intake
Posts: 395
Own Kudos [?]: 17011 [65]
Given Kudos: 217
Location: United Kingdom
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
Math Expert
Joined: 02 Sep 2009
Posts: 94351
Own Kudos [?]: 641007 [31]
Given Kudos: 85011
General Discussion
CEO
Joined: 24 Jul 2011
Status: World Rank #4 MBA Admissions Consultant
Posts: 3189
Own Kudos [?]: 1599 [2]
Given Kudos: 33
GMAT 1: 780 Q51 V48
GRE 1: Q170 V170
Senior Manager
Joined: 24 Mar 2010
Posts: 332
Own Kudos [?]: 200 [1]
Given Kudos: 4
Re: An infinite sequence of positive integers is called a perfect sequence [#permalink]
1
Bookmarks
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: 612
Own Kudos [?]: 654 [1]
Given Kudos: 2
Location: Cambridge, MA
Re: An infinite sequence of positive integers is called a perfect sequence [#permalink]
1
Bookmarks
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!
Tutor
Joined: 16 Oct 2010
Posts: 15110
Own Kudos [?]: 66644 [1]
Given Kudos: 436
Location: Pune, India
Re: An infinite sequence of positive integers is called a perfect sequence [#permalink]
1
Kudos
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"
Intern
Joined: 05 Jun 2012
Posts: 33
Own Kudos [?]: 55 [0]
Given Kudos: 66
Schools: IIMA
Re: An infinite sequence of positive integers is called a perfect sequence [#permalink]
Indeed a very nice question !!!

Thanks again Bunnel for presenting solution with such a simplicity!!!
Manager
Joined: 12 Mar 2023
Posts: 248
Own Kudos [?]: 99 [0]
Given Kudos: 16
Location: India
An infinite sequence of positive integers is called a perfect sequence [#permalink]
Bunuel 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 --> 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: https://gmatclub.com/forum/what-is-the- ... 26635.html

Hope it helps.

in statement 1 it is said that exactly one element is a prime number. There is no information about the rest of the numbers then how can take the set as prime? it is not sufficient ? i am i right?
Math Expert
Joined: 02 Sep 2009
Posts: 94351
Own Kudos [?]: 641007 [0]
Given Kudos: 85011
An infinite sequence of positive integers is called a perfect sequence [#permalink]
pudu wrote:
Bunuel 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 --> 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: https://gmatclub.com/forum/what-is-the- ... 26635.html

Hope it helps.

in statement 1 it is said that exactly one element is a prime number. There is no information about the rest of the numbers then how can take the set as prime? it is not sufficient ? i am i right?

For a sequence of positive integers to be a perfect sequence, it must consist exclusively of perfect numbers. Statement (1) indicates that one of the terms in sequence S is a prime number. As prime numbers are not perfect numbers, this means that not all terms in sequence S are perfect numbers. Consequently, sequence S does not meet the criteria for being a perfect sequence.

Hope it's clear.
Non-Human User
Joined: 09 Sep 2013
Posts: 33979
Own Kudos [?]: 851 [0]
Given Kudos: 0
Re: An infinite sequence of positive integers is called a perfect sequence [#permalink]
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]
Moderator:
Math Expert
94349 posts