The following are originally developped by Marty Tippens and used per his permission.

Math 094 - Mod. II - Chapter 4

Greatest Common Factor ( see Module II, page 29)

The procedure for finding the GCF is as follows:

1. Find the prime factorization of each number (use prime factor tree method in Ch. 3 )

2. Identify COMMON primes.

3. Pull the common primes with the least power.

4. The product of step 3 is the GCF .

Go over the examples in the module . You will be finding GCF's and LCM's from now through the end of the semester .

Example: The following is an example that uses the tree method .

...........................................

From the tree we see that the prime factorization of

1248 = and

420 = .

GCF =

(The primes common to both numbers are 2 and 3 . But 2 is not the GCF! We need to take the least exponent attached to a 2 , in this case that means 2 . Thus is the GCF of 1248 and 420.)

Note: With two numbers as large as 420 and 1248 is may seem strange that the biggest number that divides evenly into both numbers is 4.But what about 1248 and 1249 ? The biggest number that divides evenly into both of these numbers is 1 ! Why ?

 

Least Common Multiple (Module II, page 31)

The procedure for finding the LCM is as follows:

1. Find the prime factorization of each number (use prime factor tree method )

2. List ALL primes used in either of the numbers .

3. Attach the highest exponents belonging to each given prime .

4. The product of these primes with highest exponents is the LCM.

.

Example: To find the LCM of 1248 and 420 find the prime factors as follows :

...

Thus the prime factorization of

1248 = and

420 =

LCM =

(Following the procedure to step 2 we take ALL primes in each factorization and list them : 2 , 3 , 5 , 7 , 13 . These are the different numbers that were circled above. Now we take the highest exponent belonging to each prime . The highest exponent belong to 2 is 5 . The highest exponent belonging to 3, 5, 7, and 13 is 1 ( a number with no exponent actually has the "implied" exponent 1). Hence the LCM of 1248 and 420 is .

 

Practice Problem:

#1 Find the GCF of {72, 630, 378}

#2 Find the LCM of {72, 630, 378}

Solution

 

Solution

#1 Find the prime factorization of {72, 630, 378}first:

72 =

630 =

378 =

Common prime are 2 and 3. The least exponent attached to 2 is 1 and the least exponent attached to 3 is 2. Therefore,

GCF = = = 18.

#2 List ALL prime from above: 2, 3, 5, 7, then take the highest exponent attached to each prime, which is 3, 2, 1 and 1 respectively. Therefore,

LCM = . You may leave your answer in exponential form.

 

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