Factors of 81 are 1, 3, 9, 27, 81. Including 1 and 81 itself, there are 5 distinct factors for 81.
The prime factors of 81 are 3, and its factor pairs are (1, 81), (3, 27), (9, 9). We've put this below in a table for easy sharing.
| Factors of 81: | 1, 3, 9, 27, 81 |
| Prime Factors of 81: | 3 |
| Factor Pairs of 81: | (1, 81), (3, 27), (9, 9) |
How to calculate factors?
To be a factor of 81, a number must divide 81 exactly, leaving no remainder. In other words, when 81 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 81 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 81 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 81:
| Divisor | Is it a factor of 81? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 81 = 81 |
| 2 | No, 81 is not an even number so it's not divisible by 2. | - |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 27 = 81 |
| 4 | No, the last two digits (81) do not form a number divisible by 4. | - |
| 5 | No, last digit is 1, so not divisible by 5. | - |
| 6 | No, 81 is not divisible by both 2 and 3. | - |
| 7 | No, 81 divided by 7 leaves a remainder of 4. | - |
| 8 | No, the last three digits (81) do not form a number divisible by 8. | - |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 9 = 81 |
| 10 | No, last digit is 1, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (7) is not divisible by 11. | - |
| 12 | No, 81 is not divisible by both 3 and 4. | - |
| ... | 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 81
You start by dividing 81 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 \( \sqrt81 \)
| Prime Number | Is it a factor of 81? | Verification |
|---|---|---|
| 2 | No, 81 is not divisible by 2. | - |
| 3 | Yes, 81 is divisible by 3. | 81 ÷ 3 = 27, R0 |
| 3 | Yes, the result 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 81 by performing successive divisions. This method involves dividing 81 by every integer from 1 up to 81 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 81, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 81 ÷ 1 = 81 |
| 3 | 81 ÷ 3 = 27 |
| 9 | 81 ÷ 9 = 9 |
| 27 | 81 ÷ 27 = 3 |
| 81 | 81 ÷ 81 = 1 |
Using the division method, we calculated that factors of 81 are 1, 3, 9, 27, 81.
Factor Tree of 81
The factor tree of 81 shows the step-by-step breakdown of 81 into its prime factors. Each branch of the tree represents a division of 81 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 81 and highlights the structure of its prime factorization.
| 81 | |||
| | | \ | ||
| 3 | 27 | ||
| | | \ | ||
| 3 | 9 | ||
| | | \ | ||
| 3 | 3 |
Factor Pairs of 81 (Visualization)
Factor pairs of 81 are sets of two numbers that, when multiplied together, result in 81. Factor pairs are symmetric and mirror around the square root of 81, such as (1, 81) and (81, 1), and can be both positive and negative pairs as long as their product equals 81.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -81) | (1, 81) |
| (-3, -27) | (3, 27) |
| (-9, -9) | (9, 9) |
All factor pairs of 81 are (1, 81), (3, 27), (9, 9), (-1, -81), (-3, -27), (-9, -9).
Why Should I Care About Factors of 81?
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 Garage Sale: You have 81 items to sell at a garage sale. If each table can display 1 items, you’ll need 81 tables because 1 × 81 = 81.
- Filling Water Balloons: You have 81 water balloons to fill. If each group fills 3 balloons, you’ll need 27 groups because 3 × 27 = 81.
- Making Cookies: You have 81 cookie dough balls to bake. If each tray holds 9 dough balls, you’ll need 9 trays because 9 × 9 = 81.
- Arranging Books on a Shelf: You have 81 books to place on shelves. If each shelf holds 1 books, you’ll fill 81 shelves because 1 × 81 = 81.
- Setting Up a Movie Night: You have 81 snacks for movie night. If each guest gets 9 snacks, you’ll have enough for 9 guests because 9 × 9 = 81.