Hold off on the dumb brute force computer and do some actual analysis first!
So we'll first look at how the operation works on powers of primes.
a=p^i, b=p^j, c=p^k
Without loss of generality let k be the largest exponent. We will have two cases, one case when k is strictly greater than i and j, and one case when at least one of i and j equals k.
In the first case we then get
lcm(gcd(a,b),c) = c
lcm(gcd(b,c),a) = max(a,b)
lcm(gcd(c,a),b) = max(a,b)
So from this case we can conclude if we are given (with k>j)
lcm(gcd(a,b),c) = p^k
lcm(gcd(b,c),a) = p^j
lcm(gcd(c,a),b) = p^j
then we can say c is a multiple of p^k and at least one of a and b is a multiple of p^j. In the context of the main problem we can also say that the smaller of a and b is coprime to p since we are to minimize a+b+c.
In the second case we then get
lcm(gcd(a,b),c) = c
lcm(gcd(b,c),a) = c
lcm(gcd(c,a),b) = c
So from this case we can conclude if we are given
lcm(gcd(a,b),c) = p^k
lcm(gcd(b,c),a) = p^k
lcm(gcd(c,a),b) = p^k
then we can say at least two of a, b, and c are multiples of p^k. In the context of the main problem we can also say that the smallest of a, b, and c is coprime to p since we are to minimize a+b+c.
Now lets apply these to the main problem. Each of 180, 360, and 540 can be factored into 2^2*3^2*5, 2^3*3^2*5, and 2^2*3^3*5. Each of the prime bases can be treated independently of the others.
So for prime base 2 we can say a is a multiple of 2^3 and one of b or c is a multiple of 2^2.
For prime base 3 we can say b is a multiple of 3^3 and one of a and c is a multiple of 3^2.
For prime base 5 we can say two of a, b, and c are multiples of 5.
From these constraints we have a total of 12 possible sums for minimum a+b+c. From here it is pretty simple to list all 12 options and find the minimum occurs when a=2^3*1*5=40, b=2^2*3^3*1=108 and c=1*3^2*5=45 with a+b+c=189.
Analytic Solution
