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

Expert Answers
steveschoen eNotes educator| Certified Educator

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:


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


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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jdweimer eNotes educator| Certified Educator

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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rachellopez | Student

To find the GCD and LCM, I find the easiest way to understand it is to use a factor tree. Using a visual is the best way to understand a math problem in my opinion. In a factor tree you want to end with all prime numbers. I have attached a photo but here are the numbers you will end with.

18 = 2*3*2

78 = 2*3*13

198 = 2*3*2*11

For the GCD you want to find the prime numbers that all three have in common. These numbers are 2 and 3. When you multiply 2 and 3 you get a GCD of 6.

For LCM you want to multiply all of the prime numbers that are shared, which was 2, 3, 3, 11, and 13. You include 3 twice because it was shared by all three factors and also by just two factors. Your LCM would be 2574.

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parama9000 | Student



198= 2*11*3*3

Common multiple: 2*3=6

laurto | Student

To find the GCD (greatest common divisor) and the LCM (least common multiple) you have to factor. 

6 is the GCD and 2574 is the LCM 

mchandrea | Student

To get the LCM (Least common multiple) and the GCD (Greatest common divisor) You must first do prime factorization. The easiest way to do this is by using  a factor tree. I have attached a photo. 

The LCM is the smallest non-zero number that is a multiple of all the given numbers. To get it you must multiply all the prime factors that are common in all the given numbers. In this case, 

LCM = 2 * 3 * 3 * 11 * 14

LCM =  2,574

The GCD is also known as the greatest common factor. To get it you must find all prime numbers that is shared by all your given numbers. In this case, it is 2 and 3. You multiply all numbers and the product would be the GCD.

GCD = 2 * 3

GCD = 6

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