Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180. Including 1 and 180 itself, there are 18 distinct factors for 180.
The prime factors of 180 are 2, 3, 5, and its factor pairs are (1, 180), (2, 90), (3, 60), (4, 45), (5, 36), (6, 30), (9, 20), (10, 18), (12, 15). We've put this below in a table for easy sharing.
| Factors of 180: | 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 |
| Prime Factors of 180: | 2, 3, 5 |
| Factor Pairs of 180: | (1, 180), (2, 90), (3, 60), (4, 45), (5, 36), (6, 30), (9, 20), (10, 18), (12, 15) |
How to calculate factors?
To be a factor of 180, a number must divide 180 exactly, leaving no remainder. In other words, when 180 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 180 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 180 using four methods: basic factorization, prime factorization, the division method, or using GCD and LCM. We also include a detailed analysis of factor pairs and a factor tree to illustrate the breakdown.
Method 1: Basic Factorization
Basic Factorization is a method to find the factors of a number by systematically testing each whole number from 2 up to the number itself to see which ones divides with zero remainder (evenly). The process is somewhat time consuming if a number is high, that's why you should master divisibility rules, to make the process faster.
Here's the breakdown for 180:
| Divisor | Is it a factor of 180? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 180 = 180 |
| 2 | Yes, 180 is an even number so it's divisible by 2. | 2 × 90 = 180 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 60 = 180 |
| 4 | Yes, the last two digits (80) form a number divisible by 4. | 4 × 45 = 180 |
| 5 | Yes, 180 ends with 0 or 5, so it's divisible by 5. | 5 × 36 = 180 |
| 6 | Yes, 180 is divisible by both 2 and 3. | 6 × 30 = 180 |
| 7 | No, 180 divided by 7 leaves a remainder of 5. | - |
| 8 | No, the last three digits (180) do not form a number divisible by 8. | - |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 20 = 180 |
| 10 | Yes, 180 ends with 0, so it's divisible by 10. | 10 × 18 = 180 |
| 11 | No, the difference between sums of alternating digits (7) is not divisible by 11. | - |
| 12 | Yes, 180 is divisible by both 3 and 4. | 12 × 15 = 180 |
| ... | continue with all the other numbers. |
Method 2: Prime Factorization
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller natural numbers. For example:
- 2 is a prime number because its only divisors are 1 and 2.
- 3 is prime for the same reason—it can only be divided evenly by 1 and 3.
- 4 is not prime because it can be divided by 1, 2, and 4.
- 5, 7, 11, and 13 are also prime numbers.
Prime numbers are fundamental in mathematics because they are the "building blocks" of whole numbers. Any natural number greater than 1 can be expressed as a product of prime numbers, which is known as its prime factorization.
How to do prime factorization of 180
You start by dividing 180 to each prime number, multiple times, until the remainder is 0. Then you move on to the next prime number. To save time, you should test with up to \( \sqrt180 \)
| Prime Number | Is it a factor of 180? | Verification |
|---|---|---|
| 2 | Yes, 180 is divisible by 2. | 180 ÷ 2 = 90, R0 |
| 2 | Yes, the result 90 is divisible by 2. | 90 ÷ 2 = 45, R0 |
| 2 | No, the result 45 is not divisible by 2. | - |
| 3 | Yes, 45 is divisible by 3. | 45 ÷ 3 = 15, R0 |
| 3 | Yes, the result 15 is divisible by 3. | 15 ÷ 3 = 5, R0 |
| 3 | No, the result 5 is not divisible by 3. | - |
| 5 | 5 is a prime number. | 5 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 180 by performing successive divisions. This method involves dividing 180 by every integer from 1 up to 180 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 180, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 180 ÷ 1 = 180 |
| 2 | 180 ÷ 2 = 90 |
| 3 | 180 ÷ 3 = 60 |
| 4 | 180 ÷ 4 = 45 |
| 5 | 180 ÷ 5 = 36 |
| 6 | 180 ÷ 6 = 30 |
| 9 | 180 ÷ 9 = 20 |
| 10 | 180 ÷ 10 = 18 |
| 12 | 180 ÷ 12 = 15 |
| 15 | 180 ÷ 15 = 12 |
| 18 | 180 ÷ 18 = 10 |
| 20 | 180 ÷ 20 = 9 |
| 30 | 180 ÷ 30 = 6 |
| 36 | 180 ÷ 36 = 5 |
| 45 | 180 ÷ 45 = 4 |
| 60 | 180 ÷ 60 = 3 |
| 90 | 180 ÷ 90 = 2 |
| 180 | 180 ÷ 180 = 1 |
Using the division method, we calculated that factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
Factor Tree of 180
The factor tree of 180 shows the step-by-step breakdown of 180 into its prime factors. Each branch of the tree represents a division of 180 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 180 and highlights the structure of its prime factorization.
| 180 | ||||
| | | \ | |||
| 2 | 90 | |||
| | | \ | |||
| 2 | 45 | |||
| | | \ | |||
| 3 | 15 | |||
| | | \ | |||
| 3 | 5 |
Factor Pairs of 180 (Visualization)
Factor pairs of 180 are sets of two numbers that, when multiplied together, result in 180. Factor pairs are symmetric and mirror around the square root of 180, such as (1, 180) and (180, 1), and can be both positive and negative pairs as long as their product equals 180.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-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) |
All factor pairs of 180 are (1, 180), (2, 90), (3, 60), (4, 45), (5, 36), (6, 30), (9, 20), (10, 18), (12, 15), (-1, -180), (-2, -90), (-3, -60), (-4, -45), (-5, -36), (-6, -30), (-9, -20), (-10, -18), (-12, -15).
Why Should I Care About Factors of 180?
Turns out, factors aren’t just about boring math equations—they’re like secret superpowers hiding inside numbers! Knowing them can help you split things up, share with friends, or even spot hidden patterns. Want to know how? Check out these real-life examples that show just how cool factors really are:
- Building a Wall: A builder has 180 bricks and wants to build a wall. If each section of the wall uses 3 bricks, the wall will have 60 sections because 3 × 60 = 180.
- Hosting a Party: You have 180 party favors and want to give each guest the same amount. If there are 10 guests, each guest gets 18 party favors because 10 × 18 = 180.
- Making Cards: You have 180 cards to design for an event. If each person designs 2 cards, 90 people are needed because 2 × 90 = 180.
- Organizing A Car Wash: You have 180 cars to wash. If each team can wash 4 cars, you’ll need 45 teams because 4 × 45 = 180.
- Building Stairs: You have 180 steps to build for a staircase. If each section includes 2 steps, you’ll have 90 sections because 2 × 90 = 180.