How to solve system with gcd an lcm

Question

How do i solve this system in integers? $$ n+m=884 \\ {m,n}=189(m,n)$$ where n,m - lcm and m,n - gcd. I've tried using $$nm={n,m}(n,m)$$ but got stuck.

Answer 1

WLOG $\dfrac m M=\dfrac nN=(m,n)=d$(say)

$$dMN=189d\implies MN=189=7\cdot3^3$$

$$d(M+N)=884$$

Now as $(M,N)=1, (M,N)\in\{(1,189);(7,27)\}$

Now $1+189$ does not divide $884$

and $\dfrac{884}{7+27}=26\implies d=26$