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The solution and the answer by @MathLover1 is INCORRECT.
I can easily give a counter-example.
Let a= 4, b= 3 and c= 5.
Then LCM(a,b) = LCM(4,3) = 12;
LCM(b,c) = LCM(3,5) = 15 - so, the premise is satisfied;
but LCM(a,c) = LCM(4,5) = 20, which contradicts to the solution by @Mathlover1.
I can easily give another counter-example.
Let a= 12, b= 3 and c= 5.
Then LCM(a,b) = LCM(12,3) = 12;
LCM(b,c) = LCM(3,5) = 15 - so, the premise is satisfied;
but LCM(a,c) = LCM(12,5) = 60, which contradicts to the solution by @Mathlover1.
The correct solution is below.
1. The premise LCM(a,b) = 12 and LCM(b,c) = 15 implies that
c is multiple of 5 and b is not multiple of 4.
2. Then EITHER "a" is 4 OR "a" is 12,
and both/each of these two opportunities may have place.
3. It implies that LCM(a,c) is EITHER LCM(4,5) = 20
OR LCM(12,5) = 60,
and, as my counter-examples above show, each and both these opportunities may have place.
4. So, the answer to the problem's question is EITHER 20 OR 60.
Each of these two opportunities may have place.
Solved.