**Introduction to Factors and Divisibility**

*The control of large numbers is possible and like unto that of small numbers, if we subdivide them*.

—Sun Tze (544–496 B.C.)

This section will show you interesting facts about factors and factorization, including shortcuts and clues. You will also discover how to find the GCF and LCM—and what these letters stand for!

**A number is** a factor of a second number if it can be divided into the second number without leaving a remainder. Let's look at the factors of 12: 1, 2, 3, 4, 6, and 12. The number 12 can be divided by each of these numbers without there being a remainder.

12 ÷ 1 = 12 |

12 ÷ 2 = 6 |

12 ÷ 3 = 4 |

12 ÷ 4 = 3 |

12 ÷ 6 = 2 |

12 ÷ 12 = 1 |

If you start with 1 and the number itself when you write down factor pairs, you won't forget any of them. For 12, the factor pairs are as follows:

1 and 12 |

2 and 6 |

3 and 4 |

**Divisibility Shortcuts**

If a number is a factor of a given number, the given number is divisible by the factor. There are a few simple rules that will help you quickly determine divisibility and factor problems.

- An integer is divisible by 2 if its ones digit is divisible by 2.
- An integer is divisible by 3 if the sum of its digits is divisible by 3.
- An integer is divisible by 4 if its last two digits form a number divisible by 4.
- An integer is divisible by 5 if its ones digit is either 0 or 5.
- An integer is divisible by 6 if it is divisible by both 2 and 3.
- An integer is divisible by 9 if the sum of its digits is divisible by 9.
- An integer is divisible by 10 if its ones digit is 0.

**Prime Factorization**

When an integer greater than 1 has exactly two factors (1 and itself), it is a **prime number**. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that the opposite (negative version) of these numbers are also prime. For example, the factors of –23 are 1, –23, –1, and 23. Thus, –23 is prime because it has exactly two positive factors: 1 and 23.

The number 2 is the only even prime number. |

When an integer greater than 1 has more than two factors, it is called a **composite number.**

The numbers 0 and 1 are neither prime nor composite. Zero has an infinite number of factors. The number 1, on the other hand, has one factor—itself.

When a number is expressed as a product of factors that are all prime, that expression is called the **prime factorization** of the number.

**Greatest Common Factor**

The greatest of all the factors common to two or more numbers is called the **greatest common factor (GCF).**

- Let's find the common factor of 24 and 40.

- First, find the factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24.

- Now, find the factors of 40: 1, 2, 4, 5, 8, 10, 20, and 40.

- What is the greatest factor common to both 24 and 40? The greatest common factor to both is 8.

**Least Common Multiple**

A multiple of a number is the product of that number and any whole number. The least of the common multiples of two or more numbers, excluding 0, is called the **least common multiple (LCM).**

- Let's determine the least common multiple of 4, 6, and 8.

- First, find the multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, and so on.

- Now, find the multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, and so on.

- Finally, find the multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, and so on.

If you look at 4 and 6 only, you may mistakenly pick 12 as the LCM of all three numbers. But, 12 is not a multiple of 8. Instead, the first multiple found in all three numbers is 24.

Find practice problems and solutions at Factors and Divisibility Practice Questions.

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