# How do i solve this system : $\operatorname{lcm}(a,b)=8160, \gcd(a,b)=5$? with integers $a, b >5$?

let integers $a, b >5$ such that $\operatorname{lcm}(a,b)=8160, \gcd(a,b)=5$ , I have tried out to use $\operatorname{lcm} (a,b)\cdot \gcd(a,b)$ to get $a.b$ and putting $a=d a', b =db'$ with $(a',b')=1$ then $d^2\mid40800$ , then it must to set all quadratic divisors which divide $40800$ , now how do I complet the solution to determine the pair$(a',b')$ or by other method ?

$8160 = 2^5\cdot3\cdot 5\cdot 17$

$\gcd(a,b) = 5$

How can you pair the necessary factors that fit the criteria?

if both $a,b> 5$ then $a = 2^5\cdot 5\cdot 17, b = 3\cdot 5$ is one solution $a = 2^5\cdot 5, b = 3\cdot 5\cdot 17$ is another and $(a,b)=(85,480)$

• $8160$ is divisible by 17. – Matthew Conroy Jun 26 '17 at 22:54
• duh... thanks. will fix – Doug M Jun 26 '17 at 22:54
• what about (85,480) ? – zeraoulia rafik Jun 26 '17 at 23:10
• I did not mean to suggest that I had exhausted all of the combinations of prime factors. – Doug M Jun 26 '17 at 23:23
• but the question to determine all pairs , may i'm wrong – zeraoulia rafik Jun 26 '17 at 23:32