SOLUTION: The LCM of a and b is 12, and the LCM of b and c is 15. What is the value of the LCM of a and c?

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Question 1140911: The LCM of a and b is 12, and the LCM of b and c is 15. What is the value of the LCM
of a and c?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if the LCM+of a and b is 12,
the set of primes that could be a and b is:
12=1%2A2%2A2%2A3


and, if the LCM of+b and c is 15. the set of primes that could be b and c is:
15=1%2A3%2A5
What is the value of the LCM of a and c?
The LCM(a,c) is calculated by finding the prime factorization of both a and c then taking the product of the sets of primes with the highest exponent value among a and c.
from above, we know that the sets of primes of a and c could be:
a=>1%2A2%2A2%2A3
and
c=>1%2C3%2C5

both have in common 1%2C3=> so +a=1 and c=3, or+a=3 and c=1
so, the LCM+of a and+c: is+3

Answer by ikleyn(52878) About Me  (Show Source):
You can put this solution on YOUR website!
.

            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.