Factors of 3 are 1, 3. Including 1 and 3 itself, there are 2 distinct factors for 3.
The prime factors of 3 are 3, and its factor pairs are (1, 3). We've put this below in a table for easy sharing.
| Factors of 3: | 1, 3 |
| Prime Factors of 3: | 3 |
| Factor Pairs of 3: | (1, 3) |
How to calculate factors?
To be a factor of 3, a number must divide 3 exactly, leaving no remainder. In other words, when 3 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 3 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 3 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 3:
| Divisor | Is it a factor of 3? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 3 = 3 |
| 2 | No, 3 is not an even number so it's not divisible by 2. | - |
| 3 | Yes, the sum of its digits (3) is divisible by 3. | 3 × 1 = 3 |
| ... | 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 3
You start by dividing 3 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 \( \sqrt3 \)
| Prime Number | Is it a factor of 3? | Verification |
|---|---|---|
| 2 | No, 3 is not divisible by 2. | - |
| 3 | 3 is a prime number. | 3 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 3 by performing successive divisions. This method involves dividing 3 by every integer from 1 up to 3 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 3, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 3 ÷ 1 = 3 |
| 3 | 3 ÷ 3 = 1 |
Using the division method, we calculated that factors of 3 are 1, 3.
Factor Tree of 3
The factor tree of 3 shows the step-by-step breakdown of 3 into its prime factors. Each branch of the tree represents a division of 3 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 3 and highlights the structure of its prime factorization.
| 3 |
Factor Pairs of 3 (Visualization)
Factor pairs of 3 are sets of two numbers that, when multiplied together, result in 3. Factor pairs are symmetric and mirror around the square root of 3, such as (1, 3) and (3, 1), and can be both positive and negative pairs as long as their product equals 3.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -3) | (1, 3) |
All factor pairs of 3 are (1, 3), (-1, -3).
Why Should I Care About Factors of 3?
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:
- Organizing a Scavenger Hunt: You have 3 clues for a scavenger hunt. If you divide the hunt into 1 groups, each group will need 3 clues because 1 × 3 = 3.
- Building Fences: You have 3 fence posts for a garden. If each side of the garden needs 1 posts, you’ll build 3 sides because 1 × 3 = 3.
- Decorating Cupcakes: You have 3 cupcakes to decorate. If each person decorates 1 cupcakes, you’ll need 3 people to finish decorating because 1 × 3 = 3.
- Baking Brownies: You have 3 brownies to cut into pieces. If you want each person to get 1 pieces, you’ll serve 3 people because 1 × 3 = 3.
- Planting a Garden: You have 3 plants to place in a garden. If each row has 1 plants, you’ll create 3 rows because 1 × 3 = 3.