How many ordered pairs of positive integers (x, y) exist such that the

Question

How many un-ordered pairs of positive integers (x, y) exist such that the product of x, y and the greatest common factor of (x, y) is 1080?

A. 7
B. 8
C. 9
D. 10
E. 11

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GMATinsight · Accepted Answer

x*y* GCD(x,y) = 1080

But    PROPERTY : GCD (x,y) * LCM(x,y) = x*y 
Also, x = GCD*a and y = GCD*b where a and b have no common factor other than 1

i.e. GCD*a * GCD*b * GCD = 1080 = 12*9*10 = 2^3*3^3^*5

i.e.  GCD^3*a*b = 2^3*3^3*5

i.e. GCD = 1 or 2 or 3 or 6

i.e. a*b can be 1080, 135, 40 or 5 respectively

a number can be written as a product of two co-prime numbers in  2^{n-1}  ways where n is th number of Distinct prime factors of the Number

1080 (3 prime factors 2, 3, and 5) can be written as product of two coprime numbers in  2^{3-1} = 4  ways
135 (2 prime factors 3, and 5) can be written as product of two coprime numbers in  2^{2-1} = 2  ways
40 (2 prime factors 2, and 5) can be written as product of two coprime numbers in  2^{2-1} = 2  ways
5 (1 prime factors 5) can be written as product of two coprime numbers in  2^{1-1} = 1  ways

Total ways = 4+2+2+1 = 9

Answer: Option C

nick1816 · Answer

GCF(x,y) = k
x=ka
y=kb, where a and b are coprimes

ka*kb*k = 1080

k^3 * a *b = 1080

k^3 * a *b = 2^3*3^3*5

k= 1; total ordered pairs are 2*(2^2)= 8
k= 2 or 3; total ordered pairs are 2*(2^1) = 4

k = 2*3; total ordered pairs are 2*(2^0) =2

Total ordered pairs are 8+4+4+2=18

Bunuel  i guess there is a typo. It should be unordered pairs imo.

GMATinsight · Answer

Bunuel  Correct in my opinion too it should be unordered pairs.

Bunuel · Answer

___________________
Edited. Thank you.

gurmukh · Answer

What do you mean by un - ordered pair?

Posted from my mobile device

nick1816 · Answer

in which order of x and y doesn't matter. (2,5) and (5,2) will be counted as same.

gurmukh · Answer

That can be simply call pair also. Although you have given the standard definition of un-ordered pair.
Thanks

bumpbot · Answer

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How many ordered pairs of positive integers (x, y) exist such that the

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