Let x and y be positive integers such that 2(x+y)=gcd(x,y)+lcm(x,y). Find lcm(x,y)/gcd(x,y).
Let f be the gcd of x and y. Then let x=f*g and y=f*h. g and h will necessarily be coprime positive integers.
Then our given equation becomes 2*(fg+fh)=f+fgh and the expression to evaluate becomes gh.
Working on 2*(fg+fh)=f+fgh:
3f = fgh-2fg-2fh+4f
3f = f*(g-2)*(h-2)
3 = (g-2)*(h-2)
g and h can be in either order, and there is only one factorization of 3 that will keep g and h in positive integers. Then wlog g-2=1 and h-2=3.
Then g=3 and h=5 and gh=15.
Solution