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Product of Two #s

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05 Jan 2004, 22:08

Any idea

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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05 Jan 2004, 22:21

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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05 Jan 2004, 22:24

where did you get xy = 40?
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05 Jan 2004, 22:38

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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05 Jan 2004, 23:10

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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05 Jan 2004, 23:32

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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[#permalink] 05 Jan 2004, 23:32

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