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 Your Progress

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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Is the integer n odd? (1) n is divisible by 3. (2) 2n is [#permalink]

Show Tags

06 Jan 2010, 04:56

5

This post received KUDOS

Expert's post

9

This post was BOOKMARKED

Is the integer n odd ?

(1) n is divisible by 3. Clearly not sufficient. Consider n = 3 and n = 6.

(2) 2n is divisible by twice as many positive integers as n.

TIP: When odd number n is doubled, 2n has twice as many factors as n. That's because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

(1) 3 or 6. Clearly not sufficient.

(2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Sufficient.

Answer: B.

P.S. You can attach the screenshot of a question directly to the post so that everyone will see it.

Nice explanation. Didnt know abt this rule. _________________

Is the integer n odd? (1) n is divisible by 3. (2) 2n is divisible by twice as many positive integers as n.

What do you mean by the second statement.

Merging similar topics.

2n is divisible by twice as many positive integers as n, means that the # of factors of 2n is twice the # of factors of n.

For example: # of factors of 3 is two (1, and 3 itself) and the # of factors of 2*3=6 is four (1, 2, 3, and 6 itself), so the # of factors of 6=2n is twice the # of factors of 3=n.

Let us assume \(n = 3^2*5^3\) i.e. odd Total number of factors of n = (2+1)(3+1) = 12

\(2n = 2^1*3^2*5^3\) Total number of factors of n = (1+1)(2+1)(3+1) = 24 (Twice of 12 obtained before because of additional 2)

Now assume the case where n is already even:

\(n = 2^2*3^2*5^3\) i.e. odd Total number of factors of n = (2+1)(2+1)(3+1) = 36

\(2n = 2^3*3^2*5^3\) Total number of factors of n = (3+1)(2+1)(3+1) = 48 (More than 36 but not double because 3 is replaced by 4)

This is true for any prime factor. If that prime factor, p, is not in n, then p*n will have double the total number of factors. If p is already in n, the total number of factors will increase but will never double. _________________

(When even number is doubled, 2n has 1.5 more factors as n.) Sufficient.

I'm afraid the above statement is incorrect. Let's take an even number, 4. The number of factors = 3 (1, 2, 4) If the statement above is true, then the 2n (4 * 2 = 8) will have 1.5 more factors, that is 3 + 1.5 = 4.5 factors. Instead the number of factors for 8 = 4 (1, 2, 4, 8)

I tried to find out certain relationship between a even number and its double:

If an even number is E = 2^x * p^a* q^b* r^c........ 2E= 2^(x+1) * p^a* q^b* r^c........

The number of factors that 2E has more than E is

(a+1)*(b+1)* (c+1)*.........

PS: request Bunuel to check whether the observation i derived is correct.

thanx _________________

Practice Practice and practice...!!

If my reply /analysis is helpful-->please press KUDOS If there's a loophole in my analysis--> suggest measures to make it airtight.

Re: Is the integer n odd? (1) n is divisible by 3. (2) 2n is [#permalink]

Show Tags

04 May 2012, 11:40

Quote:

(2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Bunuel, could you please explain the above TIP with an example. If n=2 and n=15? Thanks.

Re: Is the integer n odd? (1) n is divisible by 3. (2) 2n is [#permalink]

Show Tags

04 May 2012, 11:56

Expert's post

pgmat wrote:

Quote:

(2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Bunuel, could you please explain the above TIP with an example. If n=2 and n=15? Thanks.

15 has 4 factors: 1, 3, 5, and 15; 15*2=30 has 4*2=8 factors: 1, 1*2=2, 3, 3*2=6, 5, 5*2=10, 15, and 15*2=30. _________________

Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

(1) 3 or 6. Clearly not sufficient.

(2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

P.S. You can attach the screenshot of a question directly to the post so that everyone will see it.

So The following doesn't happen for even numbers?

When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2. _________________

hope is a good thing, maybe the best of things. And no good thing ever dies.

Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

(1) 3 or 6. Clearly not sufficient.

(2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

P.S. You can attach the screenshot of a question directly to the post so that everyone will see it.

So The following doesn't happen for even numbers?

When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Yes it doesn't follow for even numbers. We can prove it using the formula:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers. The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

ODD Number: Finding the number of all factors of 45: \(45= 3^2*5^1\) Total number of factors of 45 including 1 and 45 itself is \(((2+1)*(1+1)=3*2=6\) factors.

Finding the number of all factors of 90: \(90=2^1*3^2*5^1\) Total number of factors of 90 including 1 and 90 itself is \((1+1)*(2+1)*(1+1)=2*3*2=12\) factors.

Note that, the odd numbers have only odd prime factors. If you double it then you introduce factor of 2^1 in the prime factorization and hence you end up multiplying by (1+1) when finding total number of factors, which therefore gets doubled.

EVEN Number: However, the same is not true for EVEN numbers. They already have prime factorization with 2^x (x>=1) and doubling that EVEN number only increments the exponent of factor 2, but not necessarily doubles the number of factors. Finding the number of all factors of 12: \(12= 2^2*3^1\) Total number of factors of 12 including 1 and 12 itself is \(((2+1)*(1+1)=3*2=6\) factors.

Finding the number of all factors of 24: \(24= 2^3*3^1\) Total number of factors of 24 including 1 and 24 itself is \(((3+1)*(1+1)=4*2=8\) factors. _________________

Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

I feel [B] is the answer. Please find below my explanation!

Statement 1. It states that n is divisible by 3. Doesn't really prove conclusively that n is odd. n could be divisible by 6. Hence not sufficient.

Statement 2. It states that 2n has twice the factors as n. Now, assuming n as odd, 2n would have the below factors {1,n,2n,2...} and n would have {1,n} as its factors. The extra factors appear due to the fact that 2 was multiplied. In case n is not a prime and has odd factors, like 3 and 7 in it, the value will double since 2 is multiplies to individual odd factors leading to double the factors again. For example if n = 3, 2n = 6, the factors double. It doesn't happen in case of n = 4. Once can similarly test the logic for complex numbers like 35.

Hence sufficient.

Hope I my explanation is satisfactory and correct!

St1: N is divisible by 3 N can be 6 (even) or 9 (odd) Hence Insufficient

St2: 2n is divisible by twice as many divisors as N if N = 3, divisors = {1,3} that is # divisors = 2 2N = 6 = {1,2,3,6} and divisors = 4 Good solution.

Lets check for an even number also. N = 6 {1,2,3,6} and #divisors = 4 2N = 12 = {1,2,3,4,6,12} and #divisors = 6 Not a good solution as #divisors for 2N is not twice #divisors for N

Hence sufficient to say that if 2n is divisible by twice as many divisors as N, then N = ODD Ans B

Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

AD/BCE

statement 1:- n is divisible by three.If n is 6 it is even and divisible by three .If n is 9 it is divisible by three. So A alone is not sufficient. AD out

statement 2:- number of factors for n = \(a^p * b^q *c^r\) ....where a,b,c.... are prime numbers and p,q,r.... are positive integers is given by\((p+1)*(q+1)*(r+1)\).... Now 2n has twice as many positive integers as n. This is only possible if n is odd.Why??IF n is odd then n doesn't have a 2 .But for 2n n will have a 2 and we will get (1+1) in our formula which doubles the number of factors.But if n is even then we will already have 2 .Lets say the number if 2's is p.The for 2n we will have p+1 2's which doesn't double the number of factors. So n must be odd for this condition to be true.So statement 2 is sufficient.

B is the answer _________________

"Kudos" will help me a lot!!!!!!Please donate some!!!

Completed Official Quant Review OG - Quant

In Progress Official Verbal Review OG 13th ed MGMAT IR AWA Structure

Yet to do 100 700+ SC questions MR Verbal MR Quant

1. is clearly insufficient. n could be zero for instance and 0 is both even and a multiple of 3.

2. I figured this out pretty quickly, I started off visualizing n as a prime number because prime numbers have 2 factors. Moreover 2 is a prime number itself and has two factors as well, then I tried to visualize the same thing assuming n to be an odd number. Generally n must not be an even number because any even number will cause our "twice as many positive integer" rule to fall.

since we are talking about factors: squares of prime numbers have 3 divisors -the prime number itself, its square, and one.

Sufficient. _________________

learn the rules of the game, then play better than anyone else.

Re: Is the integer n odd? (1) n is divisible by 3. (2) 2n is [#permalink]

Show Tags

27 Jul 2015, 20:23

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. _________________

Excellent posts dLo saw your blog too..!! Man .. you have got some writing skills. And Just to make an argument = You had such an amazing resume ; i am glad...

So Much $$$ Business school costs a lot. This is obvious, whether you are a full-ride scholarship student or are paying fully out-of-pocket. Aside from the (constantly rising)...

They say you get better at doing something by doing it. then doing it again ... and again ... and again, and you keep doing it until one day you look...