When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200. How many different values could be the greatest common divisor of the two integers?

So $gcd(a,b)*lcm(a,b)=200$ and I know that $200=2^3*5^2$. I did a little scratch work and I'm thinking the answer is $24$. Any hints are greatly appreciated.

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    $\begingroup$ So if GCD=1 what are the possibilities, and then for other options?? $\endgroup$ – Mark Bennet Nov 11 '17 at 20:54

Let the numbers be $a,b$. The product of the GCD$(a,b)$ and LCM$(a,b)$ is $ab$. To have something be a factor of the GCD you have to have it be a factor of both $a,b$, so its square must be a factor of $200$. The only numbers whose square is a factor of $200$ are $1,2,5,10$, so those are the only GCDs possible.

  • $\begingroup$ Why is their product $ab$? $\endgroup$ – ddswsd Nov 11 '17 at 21:30
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    $\begingroup$ It is a standard result. Let $c=\gcd(a,b)$ and write $a=ce,b=cf$ $\operatorname{lcm}(a,b)=cef, \operatorname{lcm}(a,b)\gcd(a,b)=c^2ef=ab$ $\endgroup$ – Ross Millikan Nov 11 '17 at 21:36
  • $\begingroup$ Wow! That is so simple and clever! It's like I knew that without ever seeing it lol $\endgroup$ – ddswsd Nov 11 '17 at 21:48

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