What is the Prime Factorization of 48 and How Is It Shown?
The concept of factors of 48 is important in mathematics, helping students solve problems quickly and improve their foundation for topics like HCF, LCM, and prime factorization. Recognizing the factors of 48 will also aid in exam preparation, mental math, and real-life calculations.
What Are Factors of 48?
A factor of 48 is any whole number that divides 48 exactly without leaving a remainder. Factors are part of many maths topics such as divisibility, prime factorization, and number properties. Understanding factors of 48 is useful for solving HCF, LCM, and even algebraic factorization questions.
List of All Factors of 48
The complete list of factors of 48 (positive integers) is:
- 1
- 2
- 3
- 4
- 6
- 8
- 12
- 16
- 24
- 48
These are the numbers by which 48 can be divided exactly. For example, 48 ÷ 8 = 6 (no remainder), so both 8 and 6 are factors of 48.
Prime Factorization of 48
The prime factorization of 48 expresses 48 as a product of prime numbers. Let’s write the steps below:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number
So, the prime factors of 48 are: 2 × 2 × 2 × 2 × 3 = 24 × 3
This process is also shown using a factor tree:
| Step | Breakdown |
|---|---|
| 1 | 48 = 2 × 24 |
| 2 | 24 = 2 × 12 |
| 3 | 12 = 2 × 6 |
| 4 | 6 = 2 × 3 |
| 5 | Prime numbers: 2, 2, 2, 2, 3 |
Pair Factors of 48
Pair factors are two numbers whose product is 48. The pair factors of 48 are helpful for multiplication and division problems:
| Factor 1 | Factor 2 |
|---|---|
| 1 | 48 |
| 2 | 24 |
| 3 | 16 |
| 4 | 12 |
| 6 | 8 |
Remember, negative pairs (like -1 × -48 = 48) also exist, but usually only positive factors are listed unless the question asks for all integers.
Properties and Quick Facts About Factors of 48
- Factors of 48 are both even (like 2, 4, 6, 8, 12, 16, 24, 48) and odd (1, 3).
- 48 is a composite number since it has more than two factors.
- The sum of all positive factors of 48 is 124.
- All factors of 48 are divisors of 48 (they divide 48 exactly).
The concept of factors of 48 is widely used in HCF, LCM, and number property questions for school and competitive exams.
Step-by-Step: How to Find All Factors of 48
Follow these steps to list all factors of 48:
1. Write 1 × 48 (since 1 and 48 multiply to 48).2. 2 × 24 (2 divides 48 exactly).
3. 3 × 16
4. 4 × 12
5. 6 × 8
6. Write the numbers in order to list all: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Speed Trick: Fast Listing of Factors
For quick exams, check divisibility by all numbers up to the square root of 48 (~7). For each divisor, write its pair. If you divide 48 by a number and get a whole number, both are factors.
Example: 48 ÷ 8 = 6. So, both 6 and 8 are factors.
Vedantu’s live classes show such quick tricks to speed up factor-finding!
Related Topics to Practice
Try These Yourself
- List all even factors of 48.
- Which prime numbers are factors of 48?
- Find a pair of factors of 48 whose sum is 14.
- Check if 24 is a factor of 48.
Common Mistakes to Avoid
- Missing factors by skipping numbers during division.
- Not matching each factor with a pair (for example, listing 6 but forgetting 8).
- Counting duplicates or listing negative factors unless asked.
Summary Table: All You Need!
| Type | Value |
|---|---|
| Total positive factors | 10 |
| List of factors | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| Prime factorization | 24 × 3 |
| Pair factors | (1,48), (2,24), (3,16), (4,12), (6,8) |
| Sum of factors | 124 |
Classroom Tip
To quickly check if a number is a factor of 48, divide 48 by that number. If the result is a whole number, it is a factor. Vedantu’s expert teachers recommend practicing such checks with similar numbers for a strong understanding.
We covered the factors of 48 from definition to prime factorization, pair factors, and easy shortcuts. Keep practicing with Vedantu's resources and related topics to strengthen your basics in Maths and prepare for exams confidently!
FAQs on Factors of 48 Explained with List, Pair Factors & Prime Factorization
1. What are all the factors of 48?
The factors of 48 are the whole numbers that divide 48 without leaving a remainder. These are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. They represent all the numbers that can be multiplied with another whole number to equal 48.
2. What is the prime factorization of 48?
The prime factorization of 48 shows 48 expressed as a product of its prime factors. A prime number is a number greater than 1 that has only two factors: 1 and itself. The prime factorization of 48 is 24 × 3, or 2 × 2 × 2 × 2 × 3.
3. How many factors does 48 have?
The number 48 has a total of 10 positive factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. It also has 10 negative factors (-1, -2, -3, etc.).
4. What are the factor pairs of 48?
Factor pairs of 48 are sets of two numbers that multiply to 48. The positive factor pairs are: (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8). There are also corresponding negative factor pairs (-1,-48), (-2,-24) etc.
5. Is 48 a prime or composite number?
48 is a composite number. A composite number is a whole number greater than 1 that has more than two factors. Since 48 has many factors (10 positive factors, to be exact), it's not prime.
6. What are the even factors of 48?
All the factors of 48 except 1 and 3 are even. The even factors of 48 are: 2, 4, 6, 8, 12, 16, 24, and 48.
7. How do I find all the factors of 48?
To find all factors, systematically divide 48 by each whole number, starting from 1, and check if the result is also a whole number. If it is, both the divisor and the quotient are factors. Continue this process until the quotient becomes less than the divisor.
8. What is a factor tree for 48?
A factor tree visually represents the prime factorization. One example for 48 is starting with 2 x 24, then breaking down 24 into 2 x 12, then 12 into 2 x 6, and finally 6 into 2 x 3. This shows the prime factorization 2 x 2 x 2 x 2 x 3 = 24 x 3.
9. How are factors of 48 used in finding the Highest Common Factor (HCF)?
To find the HCF of 48 and another number, list the factors of both numbers. The highest number that appears in both lists is the HCF. For example, to find the HCF of 48 and 36, list the factors of both and find the largest common one.
10. How are factors of 48 used in finding the Lowest Common Multiple (LCM)?
The LCM is found using the prime factorizations of the numbers. Find the prime factorization of 48 and the other number. The LCM includes the highest power of each prime factor present in either factorization.
11. Which numbers multiply to give 48?
Many number pairs multiply to 48. These pairs are the factor pairs listed above (1,48), (2,24), (3,16), (4,12), (6,8) and their negative counterparts. You can also have combinations of more than two numbers whose product is 48 (e.g., 2 x 2 x 2 x 6).
12. Can negative numbers be factors of 48?
Yes, negative numbers can also be factors. For every positive factor of 48, there's a corresponding negative factor. For instance, if 2 is a factor (because 2 x 24 = 48), then -2 is also a factor (because -2 x -24 = 48).
