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# The greatest common factor of two positive integers is 12 and their pr

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Re: The greatest common factor of two positive integers is 12 and their pr [#permalink]
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If (2^3 *12 and 3^3*12) 12 and ( 12 and 12*2^3*3^3)
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Re: The greatest common factor of two positive integers is 12 and their pr [#permalink]
Bunuel wrote:
The greatest common factor of two positive integers is 12 and their product is 31104. How many pairs of such numbers are possible?

A. 6
B. 5
C. 3
D. 2
E. 1

Are You Up For the Challenge: 700 Level Questions

Given: The greatest common factor of two positive integers is 12 and their product is 31104.

Asked: How many pairs of such numbers are possible?

Let the pair of positive integers be (x,y) and gcd & lcm be g & l respectively

xy = 12 * l = 31104 = 2^2*3 * l = 2^7*3^5
l = 2^5*3^4

Let x be 2^a*3^b & y be 2^c*3^d
min (a, c) = 2; min (b, d) = 1
max (a, c) = 7; max(b, d) = 5

(a, c) = {(2,7),(7,2)}
(b, d) = {(1,5),(5,1)}

(x,y)= {(2^2*3^1,2^7,*3^5),(2^2*3^5,2^7,*3^1),(2^7*3^1,2^2,*3^5),(2^7,3^5,2^2*3^1)}

There are 4 such ordered pairs possible.

If order is not important, as may be the case here, (2^2*3^1 & 2^7*3^5) & (2^2*3^5 & 2^7*3^1) are possible

IMO D
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Re: The greatest common factor of two positive integers is 12 and their pr [#permalink]
Kinshook wrote:
Bunuel wrote:
The greatest common factor of two positive integers is 12 and their product is 31104. How many pairs of such numbers are possible?

A. 6
B. 5
C. 3
D. 2
E. 1

Are You Up For the Challenge: 700 Level Questions

Given: The greatest common factor of two positive integers is 12 and their product is 31104.

Asked: How many pairs of such numbers are possible?

Let the pair of positive integers be (x,y) and gcd & lcm be g & l respectively

xy = 12 * l = 31104 = 2^2*3 * l = 2^7*3^5
l = 2^5*3^4

Let x be 2^a*3^b & y be 2^c*3^d
min (a, c) = 2; min (b, d) = 1
max (a, c) = 7; max(b, d) = 5

(a, c) = {(2,7),(7,2)}
(b, d) = {(1,5),(5,1)}

(x,y)= {(2^2*3^1,2^7,*3^5),(2^2*3^5,2^7,*3^1),(2^7*3^1,2^2,*3^5),(2^7,3^5,2^2*3^1)}

There are 4 such ordered pairs possible.

If order is not important, as may be the case here, (2^2*3^1 & 2^7*3^5) & (2^2*3^5 & 2^7*3^1) are possible

IMO D

Shouldn't the max(a,c) be 5 and max(b,d) be 4 as the LCM is 2^5*3^4, and thus is the answer not only 1 pair. Experts kindly help.
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Re: The greatest common factor of two positive integers is 12 and their pr [#permalink]
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Re: The greatest common factor of two positive integers is 12 and their pr [#permalink]
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