January 20, 2019 January 20, 2019 07:00 AM PST 07:00 AM PST Get personalized insights on how to achieve your Target Quant Score. January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 52294

The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 05:50
Question Stats:
43% (02:53) correct 57% (02:45) wrong based on 455 sessions
HideShow timer Statistics




Math Expert
Joined: 02 Sep 2009
Posts: 52294

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 05:50
SOLUTIONThe function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b?A. 12 B. 16 C. 20 D. 30 E. 36 \(f(b)*g(a) = 0\): any positive integer has at least one odd factor: 1. Thus, g(a) cannot be 0, which implies that f(b) = 0. This on the other hand means that b is an odd integer (odd integers does not have even factors). \(f(a) = 1\): \(a\) has 1 even factor. This means that \(a\) must be 2 (the only positive integer which has only 1 even factor is 2). The least common multiple of \(a=2\) and \(b=odd\) is \(2*odd\). Only option D can be represented this way: \(30=2*15\). Answer: D. Kudos points given to correct solutions above.Try NEW divisibility DS question.
_________________
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




Intern
Joined: 23 Sep 2012
Posts: 20
Concentration: Technology, Operations
GPA: 4
WE: Information Technology (Computer Software)

The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
Updated on: 19 Aug 2014, 06:36
Lets begin with f(a) =1 which mean the number of even factor is 1. The only number which has 1 even factor is 2 (1 and 2). 4 has 2 even factors 2 & 4 6 and 2 even factors  2 & 6 8 has 3 even factors  2,4 & 8. Hence, we know that a=2 Now f(b).g(a)=0 if a=2, g(a)=1( number of odd factors, 2 has only 1 odd factor which is 1). now if g(a)=1, then f(b) must be 0, which means b is a number with no even factors, this can happen for 1,3,5,7,9,11,13,15,17  basically all odd numbers If we now look at the answer choices 30 is the only number which can be an factored into 2 and an odd number (15). Hence correct answer is D
Originally posted by romitsn on 10 Jun 2014, 07:51.
Last edited by romitsn on 19 Aug 2014, 06:36, edited 1 time in total.




Manager
Joined: 21 Sep 2012
Posts: 214
Location: United States
Concentration: Finance, Economics
GPA: 4
WE: General Management (Consumer Products)

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 06:18
Bunuel wrote: The function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b? A. 12 B. 16 C. 20 D. 30 E. 36 Kudos for a correct solution. Answer : a=2 and b=3 satisfy both the given conditions. f(b)*g(a)=0 Even number of factors of (3)*Number of odd factors of (2)=0*1=0 & f(a)=f(2)= Number of even factors of 2=1 Least Common Multiple of 2 & 3 from given options=12 Hence Ans=A



Math Expert
Joined: 02 Sep 2009
Posts: 52294

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 06:19



Intern
Joined: 09 Jun 2014
Posts: 3
WE: Information Technology (Investment Banking)

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 06:24
Answer D a=2, b=15 satisfy both equations derived these values by factoring answer options 12=4*3 16=4*4 20=4*5 30=2*15 36=2*2*3*3



Senior Manager
Joined: 13 Jun 2013
Posts: 275

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
10 Jun 2014, 07:53
Bunuel wrote: The function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b? A. 12 B. 16 C. 20 D. 30 E. 36 Kudos for a correct solution. f(x)= even factors g(x) = odd factors now f(a)=1 ; 2 is the only such number which satisfy this condition. hence a=2 g(a)=g(2)=1 as 2 has only one odd factor which is number 1 itself. this means f(b) must be zero. i.e. b must be odd. now the highest power of 2 in a and b is 1, therefore L.C.M must contain 1 as the highest power of 2 i.e. 2^1. now coming to the options only option D satisfy this condition hence answer must be D



Intern
Joined: 22 Apr 2014
Posts: 11
Location: India

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
11 Jun 2014, 20:27
f(x)=number of even factors of x,henceforth,f(b)=number of even factors of b and f(a)=number of even factors of a
g(x)=number of odd factors of x,henceforth,g(a)=number of odd factors of a,
f(b).g(a)=0 number of even factors of b .number of odd factors of a=0 it means either the number of even factors of b or the number of odd factors of a is 0 but all the integers at least contain one odd integer 1 as factors,so it indicates number of even factors of b is 0,which means b doesnot contain any 2 in its prime factorization
f(a)=1 it means a contains only one 2 in its prime factorization (for example a could be 2,6,10,14,18…….)
As per the definition of LCM,it consists of highest power of all the factors of two or more integers therefore,the LCM will contain only single power of 2(2^1 )
among all the answer choice only D contains single power of 2,therefore,D is the correct answer



Math Expert
Joined: 02 Sep 2009
Posts: 52294

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
12 Jun 2014, 06:47
SOLUTIONThe function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b?A. 12 B. 16 C. 20 D. 30 E. 36 \(f(b)*g(a) = 0\): any positive integer has at least one odd factor: 1. Thus, g(a) cannot be 0, which implies that f(b) = 0. This on the other hand means that b is an odd integer (odd integers does not have even factors). \(f(a) = 1\): \(a\) has 1 even factor. This means that \(a\) must be 2 (the only positive integer which has only 1 even factor is 2). The least common multiple of \(a=2\) and \(b=odd\) is \(2*odd\). Only option D can be represented this way: \(30=2*15\). Answer: D. Kudos points given to correct solutions above.Try NEW divisibility DS question.
_________________
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: 18 Aug 2006
Posts: 87
Location: United States
WE: Consulting (Telecommunications)

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
04 Sep 2015, 13:08
Bunuel wrote: SOLUTION
. This means that \(a\) must be 2 (the only positive integer which has only 1 even factor is 2).
Need not be, no? Even 6 has only one even factor, so does 10, 14, 18?



Math Expert
Joined: 02 Sep 2009
Posts: 52294

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
05 Sep 2015, 03:06



Manager
Joined: 18 Aug 2006
Posts: 87
Location: United States
WE: Consulting (Telecommunications)

Re: The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
05 Sep 2015, 11:02
Bunuel wrote: anilisanil wrote: Bunuel wrote: SOLUTION
. This means that \(a\) must be 2 (the only positive integer which has only 1 even factor is 2).
Need not be, no? Even 6 has only one even factor, so does 10, 14, 18? 6 has two even factors 2 and 6. Awww! I can't believe I thought otherwise! Thanks



Senior Manager
Joined: 02 Apr 2014
Posts: 474

The function f(x) is defined for all positive integers x as
[#permalink]
Show Tags
09 Jan 2018, 11:04
Given:
f(a) = 1 => a has one even factor and that is 2 and it is must be of form 2 * m (where m is any odd integer) f(b) * g(a) = 0 => f(b) = 0 OR g(a) = 0 => but g(a) = 0 means no odd factor, this cannot be so, as every integer 1 as odd factor, so g(x) >= 1 => so f(b) = 0 => b is odd number
so a : 2 * odd b : odd
LCM (a, b) must have only one 2. Only D has single 2 as its factor
Answer (D)




The function f(x) is defined for all positive integers x as &nbs
[#permalink]
09 Jan 2018, 11:04






