   # Finding the GCF and LCM

 Our Pre-Algebra tutorial software program contains over 60 topic areas. One of them is Finding the GCF and LCM, and this page summarizes the main ideas of this topic. This page is intended for review, and is not a substitute for the interactive, self-paced tutorials of the MathTutor pre-algebra software program.

General Explanation

When dealing with whole numbers and factoring, you must learn about the greatest common factor (GCF).

The greatest common factor of two or more numbers is the greatest number which is a factor of all the numbers.

For example, the GCF of 20 and 24 is 4. 4 is a factor of both 20 and 24, and there is no number larger than 4 that is a factor of both 20 and 24.

When finding the GCF, you must make sure you choose the greatest number that is a factor of all the numbers. To do this, multiply together the prime factors that are common to the set of numbers. Use the following steps:

1. First, find the prime factorization for each number in the set.
2. Then, compare them to find the prime factors that are common to all the numbers.
3. Multiply these factors together to find the GCF.
4. Note: if a prime number occurs more than once in each number of the set, then you must include it more than once in calculating the GCF.
The opposite of the GCF is the least common multiple (LCM). The LCM of a group of numbers is the smallest whole number that is a multiple of all the numbers. For example, the LCM of 6 and 8 is 24.

The find the LCM, first obtain the prime factorization for each number. Then multiply the factors, being sure to use each factor the greatest number of times it appears in each factorization.

The LCM of a set of numbers will never be larger than the product of those numbers, although it will often be smaller.

When working with a set of numbers, remember these hints:

• The GCF is never GREATER than the SMALLEST number of the set.
• The LCM is never LESS than the LARGEST number of the set.
• The LCM is never GREATER than the product of the numbers in the set.
Sample Problem 1

Find the GCF of 90 and 105.

Solution

First we obtain the prime factorization for each number:
```90 = 2 x 3 x 3 x 5
105 = 3 x 5 x 7
```
The common primes are 3 and 5, so the GCF = 3 x 5 = 15.

Sample Problem 2

Find the LCM of 18 and 20.

Solution

First we obtain the prime factorization for each number:
```18 = 2 x 3 x 3
20 = 2 x 2 x 5
```
2 appears twice in the second factorization. 3 appears twice in the first factorization. And 5 appears once in the second factorization. So the LCM will be the product of the numbers 2, 2, 3, 3, and 5.
```LCM = 2 x 2 x 3 x 3 x 5 = 180
```
The LCM of 18 and 20 is 180.