Factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. Including 1 and 108 itself, there are 12 distinct factors for 108.
The prime factors of 108 are 2, 3, and its factor pairs are (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), (9, 12). We've put this below in a table for easy sharing.
| Factors of 108: | 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 |
| Prime Factors of 108: | 2, 3 |
| Factor Pairs of 108: | (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), (9, 12) |
How to calculate factors?
To be a factor of 108, a number must divide 108 exactly, leaving no remainder. In other words, when 108 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 108 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 108 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 108:
| Divisor | Is it a factor of 108? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 108 = 108 |
| 2 | Yes, 108 is an even number so it's divisible by 2. | 2 × 54 = 108 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 36 = 108 |
| 4 | Yes, the last two digits (8) form a number divisible by 4. | 4 × 27 = 108 |
| 5 | No, last digit is 8, so not divisible by 5. | - |
| 6 | Yes, 108 is divisible by both 2 and 3. | 6 × 18 = 108 |
| 7 | No, 108 divided by 7 leaves a remainder of 3. | - |
| 8 | No, the last three digits (108) do not form a number divisible by 8. | - |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 12 = 108 |
| 10 | No, last digit is 8, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (9) is not divisible by 11. | - |
| 12 | Yes, 108 is divisible by both 3 and 4. | 12 × 9 = 108 |
| ... | 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 108
You start by dividing 108 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 \( \sqrt108 \)
| Prime Number | Is it a factor of 108? | Verification |
|---|---|---|
| 2 | Yes, 108 is divisible by 2. | 108 ÷ 2 = 54, R0 |
| 2 | Yes, the result 54 is divisible by 2. | 54 ÷ 2 = 27, R0 |
| 2 | No, the result 27 is not divisible by 2. | - |
| 3 | Yes, 27 is divisible by 3. | 27 ÷ 3 = 9, R0 |
| 3 | Yes, the result 9 is divisible by 3. | 9 ÷ 3 = 3, R0 |
| 3 | Yes, the result 3 is divisible by 3. | 3 ÷ 3 = 1, R0 |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 108 by performing successive divisions. This method involves dividing 108 by every integer from 1 up to 108 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 108, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 108 ÷ 1 = 108 |
| 2 | 108 ÷ 2 = 54 |
| 3 | 108 ÷ 3 = 36 |
| 4 | 108 ÷ 4 = 27 |
| 6 | 108 ÷ 6 = 18 |
| 9 | 108 ÷ 9 = 12 |
| 12 | 108 ÷ 12 = 9 |
| 18 | 108 ÷ 18 = 6 |
| 27 | 108 ÷ 27 = 4 |
| 36 | 108 ÷ 36 = 3 |
| 54 | 108 ÷ 54 = 2 |
| 108 | 108 ÷ 108 = 1 |
Using the division method, we calculated that factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.
Factor Tree of 108
The factor tree of 108 shows the step-by-step breakdown of 108 into its prime factors. Each branch of the tree represents a division of 108 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 108 and highlights the structure of its prime factorization.
| 108 | ||||
| | | \ | |||
| 2 | 54 | |||
| | | \ | |||
| 2 | 27 | |||
| | | \ | |||
| 3 | 9 | |||
| | | \ | |||
| 3 | 3 |
Factor Pairs of 108 (Visualization)
Factor pairs of 108 are sets of two numbers that, when multiplied together, result in 108. Factor pairs are symmetric and mirror around the square root of 108, such as (1, 108) and (108, 1), and can be both positive and negative pairs as long as their product equals 108.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -108) | (1, 108) |
| (-2, -54) | (2, 54) |
| (-3, -36) | (3, 36) |
| (-4, -27) | (4, 27) |
| (-6, -18) | (6, 18) |
| (-9, -12) | (9, 12) |
All factor pairs of 108 are (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), (9, 12), (-1, -108), (-2, -54), (-3, -36), (-4, -27), (-6, -18), (-9, -12).
Why Should I Care About Factors of 108?
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:
- Packaging Items: You have 108 items that need to be packed in boxes. If each box can hold 3 items, then you will need 36 boxes because 3 × 36 = 108.
- Making Cards: You have 108 cards to design for an event. If each person designs 1 cards, 108 people are needed because 1 × 108 = 108.
- Hosting a Bake Sale: You have 108 cupcakes to sell at a bake sale. If each person buys 6 cupcakes, you’ll serve 18 customers because 6 × 18 = 108.
- Distributing Newspapers: You have 108 newspapers to deliver. If each route delivers 6 newspapers, you’ll have 18 routes because 6 × 18 = 108.
- Creating a Puzzle: You have 108 pieces to make a puzzle. If each section has 4 pieces, you’ll have 27 sections because 4 × 27 = 108.