SOLUTION: The L.C.M of two numbers is 120 and their G.C.F is 6. one of the numbers is 30, what is the other number?

Easy way:

The product of the GCF and the LCM of two numbers is the product of the two numbers

Let the other number be N.

Then GCF×LCM = 30N or

      6(120) = 30N
         720 = 30N
          24 = N

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HARDER WAY but involves thinking not a memorized formula:
    

 30 = 2    ×3×5
  ? = ?  
---------------
      2    ×3   = 6
      2×2×2×3×5 = 120



Rule for L.C.M
The L.C.M. of two positive integers must have a prime factor 
the MOST number of times which ONE of them has it as a factor. 

Ruke for G.C.F
The G.C.F. of two positive integers must have a prime factor 
the LEAST number of times which BOTH of them have it as a factor.

L.C.M. = 120 has 2 as a factor three times, so the MOST number 
of times 30 or the other number has it as a factor is three times.
30 only has 2 as a factor 1 time, so the other number must have 2
as a factor 3 times.  So the other number is at least 2×2×2


G.C.F. = 6 has 2 and 3 once each as its prime factors.  30 has both these
factors once so the other number must have both of them once, so the other
number must have factor 2 and 3.

We have already determined that it must have 2 as a factor, (in fact it
must have it three times); therefore the other number has 2 as a factor 
3 times and 3 as a factor once.

So the other number is 2×2×2×3 = 24.
Infact you can fill it in from this chart:

 30 = 2    ×3×5
  ? = ?  
---------------
      2    ×3   = 6
      2×2×2×3×5 = 120

Bring up the factor not represented by one of the numbers 


 30 = 2    ×3×5
 24 = 2×2×2×3  
---------------
      2    ×3   = 6
      2×2×2×3×5 = 120


Edwin