# Basic Operations

### Topics

## Introduction to :

## Least Common Multiple (LCM)

of two or more numbers is the smallest number, other than zero, that is a multiple of each number.

## Greatest Common Factor (GCF)

of two or more numbers is the largest number that is a factor of each number.

**How to Calculate Them**

There are a few different methods we can use to find both the LCM and the GCF. We will go over each. Pick the way that makes most sense to you—in the end you should get the same answer regardless of which one you choose.

#### LCM Method 1 Example 1

Method 1 involves listing the multiples of each number, starting with the number times 1, then times 2, then times 3, and so on. When you find a multiple that both numbers share, you have found your least common multiple. Find the least common multiple of and . |

#### LCM Method 1 Example 2

Find the least common multiple of and . |

#### LCM Method 2 Example 1

The second method for finding the LCM is a little more complicated, but can take less time. To
find the LCM, list the prime factors of both numbers. Then count the number of
times each prime factor appears for each number. Multiply all prime
factors using the greatest number of times it appears in Find the least common multiple of and . |

#### LCM Method 2 Example 2

Find the least common multiple of and . |

#### GCF Method 1 Example 1

Method 1 for finding the greatest common factor starts with listing the prime factors for both numbers. Multiply all prime factors that both
numbers have in common. Be sure to include a prime factor as many times
as it appears in Find the greatest common factor of and . |

#### GCF Method 1 Example 2

Find the greatest common factor of and . |

#### GCF Method 2 Example 1

Method 2 for finding the GCF is simpler to remember, but can take more time and can lead to more mistakes, since it is easy to forget factors. To find the GCF, list all the factors of each number, not just the prime factors. It can be helpful to write these factors in pairs. Compare the factors of each number until you find the largest one that they both share. Find the greatest common factor of and . |

#### GCF Method 2 Example 2

Find the greatest common factor of and . |

#### LCM and GCF Getting Wordy Example 1

You and your dad go grocery shopping. The chocolatey Shmiff Bars that you and your three sisters like come in packages of six. How many packages should your dad buy to make sure that all four of you get an equal number of bars? |

#### LCM and GCF Getting Wordy Example 2

You are getting party favors ready for your little sister's birthday party. There are 24 lollipops, 32 stickers, and 16 mini toys. Assume that you want to use all the prizes (i.e., you don't wants any prizes left over after filling the bags). How many gift bags can you make so that there is an equal number of each prize in each bag and how many prizes will be in each bag? |

#### Exercise 1

Find the LCM of 25 and 15.

#### Exercise 2

Find the GCF of 84 and 126.

#### Exercise 3

Find the LCM of 30, 45, and 60.

#### Exercise 4

Find the GCF of 48, 24, and 60.

#### Exercise 5

You and your best friend are racing around a track. You run the track in 5 min, while friend runs it in 6 min. If you both start at the same time, how many laps will you have to run before you meet your friend back at the starting place (assuming you both run a consistent pace)?

#### Exercise 6

There are 20 girls and 16 boys in your class. Your teacher wants to make groups with a the same number of girls and boys in each group. How many students will be in each group?