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Product of Two #s

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Product of Two #s [#permalink] New post 05 Jan 2004, 21:08
Any idea :shock: Pls include reasoning along with all answer posts.

The product of two numbers is 5760 and their highest common factor is 12. How many such pairs are possible?

a. 1
b. 3
c. 2
d. 4

***My answer is (b) 3. My reasoning is as follows. Let x and y equal the two numbers.
12x * 12y = 5760
12xy = 5760
xy = 480
Possible pairs that total the product 480 are:
1*480; 2*240; 3*160, 4*120; 5*96; 6*80; 10*48; 12*40
Eliminate pairs that are not prime to one another leaves pairs:
1*480; 3*160 and 5*96

Official answer is answer choice (c) 2
Anyone?
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Pls include reasoning along with all answer posts.
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 [#permalink] New post 05 Jan 2004, 21:21
12X*12Y = 5760

X*Y = 40

factors: 1,2,4,5,8,10,20,40.


4*10, 10*4
5*8, 8*5
2*20, 20*2
40*1, 1*40

2 ??
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 [#permalink] New post 05 Jan 2004, 21:24
where did you get xy = 40?
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Pls include reasoning along with all answer posts.
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 [#permalink] New post 05 Jan 2004, 21:38
sunniboy007 wrote:
where did you get xy = 40?


12*X * 12*Y = 5760 ---> possible values:

(12*1) * (12*40) = 5760 or vice versa. (12 is the common factor.)

or, 1*40 = 5760/12*12
or, 40*1 = 5760/12*12
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 [#permalink] New post 05 Jan 2004, 22:10
I think that the only two sets of numbers with the product of 5760 and a GCF of 12 are 480*12 and 96*60.

The prime factors of 5760 are
2*2*3*2*2*2*5 X 3*2*2

The only prime that can be moved from one side of the above to the other is 5. If 3 is moved, twelve is not a factor of the first set. If a 2 is moved the GCF will increase. The first possibility above is 480*12. If the 5 moved over, the result is 96*60.

I am not sure if this makes sense in writing, but in my head it sounded ok!!
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 [#permalink] New post 05 Jan 2004, 22:32
Excellent explanation Drodger. I had it down to the prime factorization but wasn't clear about what to do next and your answer perfectly makes sense
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