Find the greatest common divisor (GCD) and the least common multiple (LCM) of 18, 78, and 198.

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Find the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of 18, 78, and 198 can be found using a few methods. The attachment illustrates a couple of ways. They both start out by finding the prime factorization of each of the numbers. For the example I used a factor tree.

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A couple of ways you can get both.  With these numbers, it would probably be easiest to use prime factorization.  Prime factorization is breaking down each number into factors of only prime numbers.  Normally, if necessarily, a factor tree is done for that.  I will show that on an attachment.

For the prime factorization:

18 = 3*3*2

78 = 2*3*13

198 = 2*3*3*11

Then, for each item being asked for:

LCM

1) Make a list of each factor from the prime factorization:

   2         3       11         13   

2) Write down the most number of times that each occurs in each factorization.  For instance, for the 2, it occurs the most in 198, six times.  So:

   2       3         11            13   

2        3*3        11           13

3) Multiply all of these together:

2*3*3*11*13 = 2574

2574 is the least common multiple

GCD

Same process as #1, except you consider how many times each factor is occurring in the prime factorizations for all three numbers.  For instance, in all prime factorizations, the "2" occurs at least once in all three of them.  "Once" means you include "one 2".  So:

   2         3       11        13   

  2          3       --         --

For example, if the 3 did occur twice in all three prime factorizations, you would include two 3's here.  The 11 and 13 don't occur in all three prime factorizations, so you don't include them.  Multiply the numbers you have:

2*3 = 6

So, 6 is the greatest common divisor.

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