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The function f(x) is defined for all positive integers x as

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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

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[Reveal] Spoiler: OA

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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

\(f(b)*g(a) = 0\): 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.

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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New post 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

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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New post 10 Jun 2014, 06:19
desaichinmay22 wrote:
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


The LCM of 2 and 3 is 6, not 12.
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Re: The function f(x) is defined for all positive integers x as [#permalink]

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New post 10 Jun 2014, 06:24
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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

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The function f(x) is defined for all positive integers x as [#permalink]

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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

Last edited by romitsn on 19 Aug 2014, 06:36, edited 1 time in total.

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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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

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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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

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SOLUTION

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

\(f(b)*g(a) = 0\): 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.
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Collection of Questions:
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Re: The function f(x) is defined for all positive integers x as [#permalink]

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New post 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?

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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Re: The function f(x) is defined for all positive integers x as [#permalink]

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New post 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

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Re: The function f(x) is defined for all positive integers x as   [#permalink] 25 Nov 2017, 03:03
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