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

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Question Stats: 45% (02:58) correct 55% (02:44) wrong based on 400 sessions

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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

Kudos for a correct solution.

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Math Expert V
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Posts: 58418
Re: The function f(x) is defined for all positive integers x as  [#permalink]

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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: $$30=2*15$$.

Kudos points given to correct solutions above.

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

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1
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, 08:51.
Last edited by romitsn on 19 Aug 2014, 07:36, 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.

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 V
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Posts: 58418
Re: The function f(x) is defined for all positive integers x as  [#permalink]

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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.

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

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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: $$30=2*15$$.

Kudos points given to correct solutions above.

Try NEW divisibility DS question.
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Re: The function f(x) is defined for all positive integers x as  [#permalink]

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

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

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

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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

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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

I did not take into account that 1 is a factor of every number, so this took me too long (4 mins) but here's the solution:

#EVEN factor a = 1 ... from this we know a = 2 since 2 is the only number with 1 even factor.

#EVEN factor b * #ODD factor a = 0 ... we know that either b has no EVEN factors or a has no ODD factors (Bunuel's solution eliminates the issue I had here with multiple cases...)

Since we know a=2 then #ODD factor a = 1, so the number #EVEN in b must be 0 so that the above equation works.

This means a=2 and b could = 3. Not a solution based on answer choices (LCM would be 6).
Since b can only have odd factors the next one is 3*5 = 15, so 2*15 = 30, which is AC D.
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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.

If f(b) = 0 ; the number of even factors of b = 0; b = odd
If g(a) = 0 ; the number of odd factors of a = 0; 1 is odd; not feasible
b = odd

f(a) = 1; the number of even factors of a = 1; a=2o where o if an odd number;

LCM(a,b) = 2o form

IMO D
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