Factors of 180 are the numbers that can be evenly divided into 180 without leaving a remainder. The factors include 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.

These factors play a crucial role in various mathematical operations and calculations, such as finding the greatest common divisor, determining prime factorization, and solving equations involving 180.

Understanding the Factors of the number 180 helps gain insights into the number’s mathematical properties and relationships with other numbers.

**What are the Factors of 180?**

The numbers can be evenly divided into 180 without leaving a remainder.

**Factors:**1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180**Negative Factors:**-1, -2, -3, -4, -5, -6, -9, -10, -12, -15, -18, -20, -30, -36, -45, —60, -90, and -180**Prime Factorization of 180:**180 = 2×2 × 3×3 × 5**The Sum of Factors:**15

180 is a highly composite number with more divisors than any smaller positive integer. The number 180 can be refactored. For a better understanding, we will calculate the Factors of number 180, their prime factors, and their factors in pairs in this lesson.

**Key Points:**

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180 are factors of 180

1 is a universal factor that affects almost all

180 has 2, 3, and 5 prime factors

**Pair Factors of 180**

The pair Factors of number 180 are the two numbers that when multiplied together, equal 180. 180 has several pair factors since it is a composite number. Below are 180’s positive and negative pair factors:

**Pair Factors of 180**

A pair of numbers multiplied together, resulting in 180, are the pair Factors of the number 180. As the number 180 is composite, it has more than one pair of factors. The positive and negative pair Factors of number 180 are given below:

**Positive Pair Factors of 180:**

Positive Factors | Positive Pair Factors |

1 × 180 | (1, 180) |

2 × 90 | (2, 90) |

3 × 60 | (3, 60) |

4 × 45 | (4, 45) |

5 × 36 | (5, 36) |

6 × 30 | (6, 30) |

9 × 20 | (9, 20) |

10 × 18 | (10, 18) |

12 × 15 | (12, 15) |

**Negative Pair Factors of 180:**

Negative Factors | Negative Pair Factors |

-1 × -180 | (-1, -180) |

-2 × -90 | (-2, -90) |

-3 × -60 | (-3, -60) |

-4 × -45 | (-4, -45) |

-5 × -36 | (-5, -36) |

-6 × -30 | (-6, -30) |

-9 × -20 | (-9, -20) |

-10 × -18 | (-10, -18) |

-12 × -15 | (-12, -15) |

**Factors of 180 by Division Method:**

The integers are the Factors of the number 180 if they divide 180 and have a remainder value of 0. The following is the division method to determine the 180 factors:

Division | Factor | Remainder |

180 ÷ 1 | 1 | 0 |

180 ÷ 2 | 2 | 0 |

180 ÷ 3 | 3 | 0 |

180 ÷ 4 | 4 | 0 |

180 ÷ 5 | 5 | 0 |

180 ÷ 6 | 6 | 0 |

180 ÷ 9 | 9 | 0 |

180 ÷ 10 | 10 | 0 |

180 ÷ 12 | 12 | 0 |

180 ÷ 15 | 15 | 0 |

180 ÷ 18 | 18 | 0 |

180 ÷ 20 | 20 | 0 |

180 ÷ 30 | 30 | 0 |

180 ÷ 36 | 36 | 0 |

180 ÷ 45 | 45 | 0 |

180 ÷ 60 | 60 | 0 |

180 ÷ 90 | 90 | 0 |

180 ÷ 180 | 180 | 0 |

**How to Find Prime Factorization of 180**

To find the prime factorization of 180, we’ll determine the prime numbers that divide it evenly.

**Step 1:** Divide by 2 repeatedly until it’s no longer divisible: 180 ÷ 2 = 90 90 ÷ 2 = 45

**Step 2:** Divide by 3 repeatedly until it’s no longer divisible: 45 ÷ 3 = 15

**Step 3:** Divide by 5 once: 15 ÷ 5 = 3

**Step 4: **3 is a prime number, so we cannot divide it** **further.

Now, let’s combine the prime factors we obtained: 180 = 2 × 2 × 3 × 3 × 5

Therefore, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5.

**Factor Tree 0f 180**

To create a factor tree for 180, we’ll break it down into its prime factors using a branching structure. Here’s how you can make a factor tree for 180:

- Start with the number 180 at the top of the tree.
- Look for the most minor prime factor of 180, which is 2. Divide 180 by 2: 180 ÷ 2 = 90
- Write 2 and 90 as branches below 180.

- Now, we’ll continue with 90. Look for its smallest prime factor, which is also 2. Divide 90 by 2: 90 ÷ 2 = 45
- Write 2 and 45 as branches below 90.

- Next, we’ll continue with 45. Look for its most minor prime factor, which is 3. Divide 45 by 3: 45 ÷ 3 = 15
- Write 3 and 15 as branches below 45.

- Now, we’ll continue with 15. Look for its most minor prime factor, which is 3. Divide 15 by 3: 15 ÷ 3 = 5
- Write 3 and 5 as branches below 15.

- Finally, we have prime numbers at the ends of the branches (2, 2, 3, 3, and 5). This represents the prime factorization of 180:

180 = 2 × 2 × 3 × 3 × 5

This factor tree illustrates how the prime factor 180 is derived through successive divisions.

**Find the common factors of 60 and 180.**

First, let’s list the factors of each number:

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factors of number 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

The common factors of 60 and 180 are:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

**Find all the Factors of number 180.**

To find the Factors of number 180, we can use the prime factorization method we discussed earlier.

The prime factorization of 180 is 2 × 2 × 3 × 3 × 5. Now, we can generate the factors by combining these prime factors in different ways:

Factors of number 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.

**What is the sum of the Factors of the number 180?**

To find the sum of the Factors of number 180, we add up all the factors:

Sum of the Factors of number 180: 1 + 2 + 3 + 4 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + 20 + 30 + 36 + 45 + 60 + 90 + 180 = 546.

Therefore, the sum of the Factors of number 180 is 546.

**How many factors does 180 have?**

To determine the number of Factors of the number 180, we count the possibilities of combining its prime factors.

The prime factorization of 180 is 2 × 2 × 3 × 3 × 5. We consider each prime factor’s exponent to find the number of factors.

The exponent of 2 is 2, the exponent of 3 is 2, and the exponent of 5 is 1.

To find the number of factors, we add 1 to each exponent and multiply them:

(2 + 1) × (2 + 1) × (1 + 1) = 3 × 3 × 2 = 18.

Therefore, 180 has 18 factors.