What are the Factors of 27? Explained with Lists & Factor Trees
The concept of factors of 27 plays a key role in mathematics and is essential for topics such as prime factorization, LCM and HCF, and many types of exam word problems. Knowing how to quickly find the factors of 27 helps students solve questions involving divisibility, multiples, and comparisons with similar numbers like 18, 36, or 45.
What Are the Factors of 27?
A factor of 27 is any integer that divides 27 completely, without leaving a remainder. In simple words, when you multiply two integers and the result is 27, then both numbers are considered factors. This idea comes up in topics like LCM, Prime Factorization, and when studying factors of a number in general. The factors of 27 are always whole numbers, and each factor is paired with another to multiply and give 27.
Key Formula for Finding Factors of 27
Here’s the standard formula: If a × b = 27, then both a and b are factors of 27. For example, 3 × 9 = 27, so both 3 and 9 are factors. The factors can also be listed by dividing 27 by each integer from 1 up to 27 and checking for zero remainder.
List of All Factors of 27
To make it simple for revision, all factors of 27 are:
- 1
- 3
- 9
- 27
Factors Table for 27
| Factor | Pair | Product |
|---|---|---|
| 1 | 27 | 1 × 27 = 27 |
| 3 | 9 | 3 × 9 = 27 |
| 9 | 3 | 9 × 3 = 27 |
| 27 | 1 | 27 × 1 = 27 |
Pair Factors of 27
Pair factors are two numbers which, when multiplied, give 27. Both positive and negative pairs are possible because (−a) × (−b) = positive product.
| Positive Pair | Negative Pair |
|---|---|
| (1, 27) | (−1, −27) |
| (3, 9) | (−3, −9) |
Prime Factorization of 27
Prime factorization means expressing 27 as a product of its prime numbers. The prime factorization of 27 uses only the number 3:
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
So, \( 27 = 3 \times 3 \times 3 = 3^3 \). The factor tree of 27 has three branches of 3.
Is 27 a Prime or Composite Number?
27 is a composite number because it has more than two factors (1, 3, 9, 27). Prime numbers have only two factors (1 and the number itself). 27 can be divided by 3 and 9 as well, so it's not a prime number.
Difference Between Factors and Multiples of 27
| Factors of 27 | Multiples of 27 |
|---|---|
| Numbers that divide 27 exactly: 1, 3, 9, 27 | Numbers you get by multiplying 27 by any whole number: 27, 54, 81, 108, ... |
How to Find Factors of 27: Step-by-Step
- Start with 1 and divide 27 by 1: remainder is 0, so 1 is a factor.
- Try 2: 27 ÷ 2 = 13.5 (not a whole number) ⇒ skip.
- Try 3: 27 ÷ 3 = 9, remainder 0 → 3 and 9 are factors.
- Keep checking up to 27. Only 1, 3, 9, and 27 divide evenly.
Speed Tricks to Remember Factors of 27
Try this: For any cube number like 27 (=3³), all the factors will also be factors of that base number (3) and its multiples. For exam speed, remember:
- If you see 27, think "cube of 3," so 1, 3, 9, and 27 come naturally.
- To check factor pairs, remember only odd numbers because 27 is odd.
Vedantu tip: Regular practice with factor tables and factor trees helps you instantly recall factors in exams!
Solved Example: Find All Factors of 27
Let’s show every step for students:
1. List all numbers from 1 to 27.2. Divide 27 by each number to see if the result is a whole number and if remainder is 0.
3. 27 ÷ 1 = 27 ✔
4. 27 ÷ 3 = 9 ✔
5. 27 ÷ 9 = 3 ✔
6. 27 ÷ 27 = 1 ✔
7. So, factors are 1, 3, 9, 27.
Applications in Word Problems
Factors of 27 are used in exam questions on LCM, HCF, and for dividing chocolates, grouping students, or working out which numbers can form exact rows. For example, when asked to find common factors of 27 and other numbers, or to split 27 objects into equal groups.
Practice Questions
- Is 9 a factor of 27? Explain why.
- List all the pair factors of 27.
- What is the prime factorization of 27?
- Which is a multiple of 27: 54 or 26?
Frequent Errors with the Factors of 27
- Forgetting that 1 and 27 itself are always factors.
- Thinking even numbers like 2 or 6 are factors (they are not, because 27 is an odd number).
- Confusing multiples with factors.
Relation to Other Maths Concepts
The factors of 27 connect with topics such as factors of 18, factors of 36, or factors of 45. Understanding factors helps you with topics like LCM, HCF, divisibility, and even algebraic factorization in higher classes.
Classroom Tip
To remember the factors of 27, visualize 27 as three rows of three 3’s (since 3 × 3 × 3 = 27), or use arrays and factor trees. In Vedantu classes, students often draw a factor tree for 27 to easily recall its prime composition: each branch ending in the number 3.
We explored factors of 27—from direct list, prime factorization, pairs, common exam errors, and important links to topics you’ll need next. Keep practicing with Vedantu to become an expert in factors and their applications for school and entrance exams!
Related Vedantu Topic Links
FAQs on Factors of 27: Definition, Pair Factors & Examples
1. What are the factors of 27?
The factors of 27 are the numbers that divide 27 exactly without leaving a remainder. These are 1, 3, 9, and 27. In other words, they are the numbers that, when multiplied together, result in 27.
2. How many factors does 27 have?
The number 27 has four factors: 1, 3, 9, and 27.
3. What is the prime factorization of 27?
The prime factorization of 27 is 3 × 3 × 3, or 33. This means 27 is expressed as a product of only prime numbers.
4. Is 27 a prime or composite number?
27 is a composite number because it has more than two factors (1 and itself).
5. What are the pair factors of 27?
The pair factors of 27 are (1, 27) and (3, 9). These are pairs of numbers that multiply to give 27.
6. What are the multiples of 27?
Multiples of 27 are numbers obtained by multiplying 27 by other whole numbers. Examples include 27, 54, 81, 108, and so on.
7. How do I find the factors of 27 using division?
To find the factors using division, divide 27 by each whole number starting from 1. If the division results in a whole number with no remainder, that number is a factor. For example: 27 ÷ 1 = 27; 27 ÷ 3 = 9; 27 ÷ 9 = 3; 27 ÷ 27 = 1.
8. What is a factor tree for 27?
A factor tree visually represents the prime factorization. For 27, the tree would show: 27 branching into 3 and 9; then 9 branching into 3 and 3. This shows 27 = 3 × 3 × 3.
9. Can negative numbers be factors of 27?
Yes, -1, -3, -9, and -27 are also considered factors of 27 because multiplying any pair of these negative factors will produce 27.
10. How are the factors of 27 used in finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?
The factors of 27 are essential when calculating the HCF and LCM with other numbers. You find the common factors to determine the HCF and use the prime factorization to find the LCM.
11. What is the difference between factors and multiples of 27?
Factors are numbers that divide 27 evenly (1, 3, 9, 27). Multiples are numbers that are the result of multiplying 27 by another whole number (27, 54, 81, etc.).
12. Are there any shortcuts to find factors quickly?
One shortcut is to check for divisibility rules (e.g., divisibility by 3). For small numbers like 27, you can quickly list the pairs that multiply to give 27. Knowing the prime factorization (33) helps find all factors efficiently.
