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

Question

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

A. 12
B. 16
C. 20
D. 30
E. 36

Bunuel · Accepted Answer

Official Solution:   
 
The function  f(x)  is defined for all positive integers  x  as the number of positive even factors of  x  and the function  g(x)  is defined for all positive integers  x  as the number of positive 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 : Every positive integer has at least one odd factor: 1. Therefore,  g(a)  cannot be 0, which implies that  f(b) = 0 . This, in turn, means that  b  is an odd integer (odd integers do not have even factors).

f(a) = 1 :  a  has one positive even factor. This means that  a  must be 2 (the only positive integer with only one positive even factor is 2).

The least common multiple of  a=2  and  b=odd  is  2*odd . Only option D could be represented this way:  30=2*15 .

Answer: D

desaichinmay22 · Answer

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

Bunuel · Answer

The LCM of 2 and 3 is 6, not 12.

shukasa · Answer

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

romitsn · Answer

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

manpreetsingh86 · Answer

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

sam26can · Answer

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

anilisanil · Answer

Need not be, no?

Even 6 has only one even factor, so does 10, 14, 18?

Bunuel · Answer

6 has two even factors 2 and 6.

hellosanthosh2k2 · Answer

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)

energetics · Answer

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.

Kinshook · Answer

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

voluptasdeserunt · Answer

Why did we not consider 2 and -2 as even factors, and 1 and -1 as odd factors.

Bunuel · Answer

I updated the wording to remove this ambiguity.

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

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