Mar 23 07:00 AM PDT  09:00 AM PDT Christina scored 760 by having clear (ability) milestones and a trackable plan to achieve the same. Attend this webinar to learn how to build trackable milestones that leverage your strengths to help you get to your target GMAT score. Mar 27 03:00 PM PDT  04:00 PM PDT Join a free live webinar and learn the winning strategy for a 700+ score on GMAT & the perfect application. Save your spot today! Wednesday, March 27th at 3 pm PST Mar 29 10:00 PM PDT  11:00 PM PDT Right now, their GMAT prep, GRE prep, and MBA admissions consulting services are up to $1,100 off. GMAT (Save up to $261): SPRINGEXTRAGMAT GRE Prep (Save up to $149): SPRINGEXTRAGRE MBA (Save up to $1,240): SPRINGEXTRAMBA Mar 30 07:00 AM PDT  09:00 AM PDT Attend this webinar and master GMAT SC in 10 days by learning how meaning and logic can help you tackle 700+ level SC questions with ease. Mar 31 07:00 AM PDT  09:00 AM PDT Attend this webinar to learn a structured approach to solve 700+ Number Properties question in less than 2 minutes.
Author 
Message 
TAGS:

Hide Tags

Senior Manager
Joined: 02 Dec 2007
Posts: 388

Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
Updated on: 04 Mar 2019, 08:09
Question Stats:
65% (01:36) correct 35% (01:41) wrong based on 841 sessions
HideShow timer Statistics
Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by Nihit on 15 Nov 2008, 09:44.
Last edited by Bunuel on 04 Mar 2019, 08:09, edited 2 times in total.
Renamed the topic, edited the question and added the OA.




Math Expert
Joined: 02 Sep 2009
Posts: 53796

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
06 Jan 2010, 04:56



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9007
Location: Pune, India

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
12 Dec 2010, 05:12
ajit257 wrote: 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. If \(n = a^p*b^q*c^r...\) where a, b, c are distinct prime numbers, Total number of factors of n = (p+1)(q+1)(r+1)... (Check out the post: http://www.veritasprep.com/blog/2010/12/quarterwitquarterwisdomwritingfactorsofanuglynumber/ for complete explanation of this formula) 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. even 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. *Edited*
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >




Intern
Joined: 14 Sep 2003
Posts: 42
Location: california

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
15 Nov 2008, 13:29
Statement #1 is not sufficient for reasons mentioned by icandy..so A and D are out as choices Statement #2 gives an interesting data point. It says 2n has exactly 2xfactors compared to n Let's take n = 6, then factors are 1, 2, 3 and 6 itself for a total count of 4 Now 2n = 12, so factors are 1, 2, 3, 4, 6, 12 for a total of 6. Another sample, say n = 8, then factors are 1, 2, 4, 8 (count 4) 2n = 16, then factors are 1, 2, 4, 8, 16 (count 5). Hmmm ok, doesn't look like even numbers satisfy this. Let's try with a couple of odd number substitutions Say, n = 5, then factors are 1, 5(count = 2) 2n = 10, factors are 1, 2, 5, 10 (count = 4)...looking good so far Say n = 9, then factors are 1, 3, 9(count = 3) 2n = 18, factors are 1, 2, 3, 6, 9, 18(count = 6). hmmm good are we ready to call n odd. Let's hold on and do a prime number test Say n = 3, then factors are 1, 3(count = 2) 2n = 6, then factors are 1, 2, 3, 6(count = 4). Good let's do the last prime # try Say n = 2, then factors are 1, 2(count = 2) 2n = 4, then factors are 1, 2, 4 (count = 3) Seems like #2 is sufficient to answer the question whether n is odd. So pick is B It'd be a better question, if it was asked whether n is a prime number? That'd have made it a little bit more tricky, by extending us to use both #1 and #2. In which case I think I'd go with C.
_________________
excellence is the gradual result of always striving to do better



Manager
Joined: 21 Jul 2009
Posts: 201
Location: New York, NY

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
18 Sep 2009, 13:29
There must be a general rule behind this to avoid plugging numbers. Anyone know?



GMAT Tutor
Joined: 24 Jun 2008
Posts: 1395

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
19 Sep 2009, 10:04
mendelay wrote: There must be a general rule behind this to avoid plugging numbers. Anyone know? There is a general rule here, which we can arrive at by extending the logic in maliyeci's excellent explanation above. I could explain this abstractly, but it's probably easier to take a specific example  let's use the number 72 = (2^3)(3^2). Now, this number has 12 factors in total, three of which are odd: 1, 3, 3^2 Now, if we multiply each of the numbers above by 2^1, we get three even divisors of 72, and the same will happen if we multiply these numbers by 2^2 or 2^3. So 72 has three odd divisors, and nine even divisors: 1, 3, 3^2 2, 2*3, 2*3^2 2^2, (2^2)*3, (2^2)(3^2) 2^3, (2^3)*3, (2^3)(3^2) Notice that we have three times as many even divisors as odd divisors because the power on the 2 in the prime factorization of 72 is 3; that guarantees that we have three even divisors for every odd divisor. You could use this logic for any number, of course, from which we have the following general rule: * The ratio of the number of even divisors of x to the number of odd divisors of x is always equal to the power on the 2 in the prime factorization of x. So, if the power on the 2 in the prime factorization of x is equal to 1, we have an equal number of odd and even divisors. If the power is greater than 1, we have more even divisors than odd divisors.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com



Math Expert
Joined: 02 Sep 2009
Posts: 53796

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
10 Dec 2010, 06:42



Intern
Joined: 26 Jul 2010
Posts: 24

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



Math Expert
Joined: 02 Sep 2009
Posts: 53796

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
04 May 2012, 11:56
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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Status: Gonna rock this time!!!
Joined: 22 Jul 2012
Posts: 440
Location: India
GMAT 1: 640 Q43 V34 GMAT 2: 630 Q47 V29
WE: Information Technology (Computer Software)

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
24 Jan 2013, 23:53
Bunuel wrote: 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. For more on this topic check Number Theory chapter of Math Book: mathnumbertheory88376.htmlP.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.
Who says you need a 700 ?Check this out : http://gmatclub.com/forum/whosaysyouneeda149706.html#p1201595
My GMAT Journey : http://gmatclub.com/forum/endofmygmatjourney149328.html#p1197992



Senior Manager
Joined: 27 Jun 2012
Posts: 365
Concentration: Strategy, Finance

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
25 Jan 2013, 00:48
Sachin9 wrote: Bunuel wrote: 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. For more on this topic check Number Theory chapter of Math Book: mathnumbertheory88376.htmlP.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.
_________________
Thanks, Prashant Ponde
Tough 700+ Level RCs: Passage1  Passage2  Passage3  Passage4  Passage5  Passage6  Passage7 Reading Comprehension notes: Click here VOTE GMAT Practice Tests: Vote Here PowerScore CR Bible  Official Guide 13 Questions Set Mapped: Click here Finance your Student loan through SoFi and get $100 referral bonus : Click here



Intern
Joined: 29 Apr 2018
Posts: 13

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
09 Nov 2018, 12:06
Bunuel wrote: ajit257 wrote: 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. Hope it's clear. Bunuel: Is this applicable only on odd numbers? Posted from my mobile device



Math Expert
Joined: 02 Sep 2009
Posts: 53796

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
10 Nov 2018, 06:11
s111 wrote: Bunuel wrote: ajit257 wrote: 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. Hope it's clear. Bunuel: Is this applicable only on odd numbers? Posted from my mobile device# of factors of 2 is two (1, and 2 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 2=n.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 10 Apr 2018
Posts: 200

Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
Show Tags
27 Feb 2019, 08:34
Hi, here are my two cents for this question Let n be expressed in terms of prime factors as \(a^{p}\) \(b^{q}\) \(c^{r}\) where a,b,c are prime numbers and p q r are postie powers of prime numbers then total number of factors of n = (p+1)(q+1)(r+1) = X then 2n if expressed in terms of its prime factors can be 2n=\(a^{p}\) \(b^{q}\) \(c^{r}\) Now if any of the prime factors of n does not contain 2 as its factor then the total number of factors of 2n would be = (1+1) (p+1)(q+1)(r+1) = 2X but if any of the prime factors of n contains 2 as its factor then the total number of factors of 2n \(\neq\) twice the number of factors of n On the same lines we can say that if number of facotrs of a number n =X, multiplying that number 'n' by another prime factor A which is not a prime factor of 'n' the total number of factors of 'An' would be 2X Let n= \(2^1 5^1\), total number of factors are 4 then 3n=\(2^1 3^1 5^1\) , total number of factors are 8 which is twice of n. n= \(2^2\), total number of factors are 3 then 3n=\(2^2 3^1\) , total number of factors are 6 which is twice of n. then 5n=\(2^2 5^1\) , total number of factors are 6 which is twice of n. So since from statement B we have that the number of factors of 2n is twice the number of factors of n we can say that 2 is not a prime factor of n. If two is not prime factor of n then n is odd. Probus
_________________
Probus




Re: Is the integer n odd? (1) n is divisible by 3 (2) 2n is divisible by
[#permalink]
27 Feb 2019, 08:34






