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When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers?

 Aug 5, 2019
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When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers?

 

 

 

Let the numbers be ax and ay   where  x and y are relatively prime and a, x, and y are all positive integers.

So the GCD is    \(a\)

and the lowest commone multiple is     axy

The product of these is      \(a^2xy\)

So we have

 

\(a^2xy=180\\ a^2 \;\;must \;\;be\;\;1,4,9, or \; 36 \;\text{no other square integerwill make this true}\\ \text{So a = 1, -1, 2, -2, 3,- 3, 6 or - 6}\\ \)

So there seems to be 8 possible values for the greatest common divisor.

 Aug 5, 2019

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