How to Find the Factors and Prime Factorization of 168?
The concept of Factors of 168 is a foundational topic in arithmetic and number theory, essential for building skills in divisibility, algebraic manipulation, and solving exam questions. Knowing how to find and use the factors of 168 helps students in topics such as HCF, LCM, and prime factorisation, making it valuable for both school and competitive exams.
What are the Factors of 168?
A factor of 168 is any integer that divides 168 exactly with no remainder. Because 168 is a composite number, it has more than just 1 and itself as factors. The complete set of positive factors of 168 is:
- 1
- 2
- 3
- 4
- 6
- 7
- 8
- 12
- 14
- 21
- 24
- 28
- 42
- 56
- 84
- 168
Negative factors are also included in higher mathematics and advanced number theory. They are simply the negatives of the positive factors, for example, -1, -2, ..., -168.
Pair Factors of 168
Pair factors are two integers whose product is 168. Pairing each factor with its match gives us:
| Positive Pair Factor | Explanation |
|---|---|
| 1 × 168 | 1 x 168 = 168 |
| 2 × 84 | 2 x 84 = 168 |
| 3 × 56 | 3 x 56 = 168 |
| 4 × 42 | 4 x 42 = 168 |
| 6 × 28 | 6 x 28 = 168 |
| 7 × 24 | 7 x 24 = 168 |
| 8 × 21 | 8 x 21 = 168 |
| 12 × 14 | 12 x 14 = 168 |
Each positive pair has a negative pair as well, since (-a) × (-b) = 168 for each (a, b) above. For example, (-2, -84), (-12, -14), etc.
How to Find the Factors of 168?
To determine all the factors of 168, use the division method—divide 168 by whole numbers, checking for results with zero remainder:
- Start with 1 and move upwards: 168 ÷ 1 = 168 (so 1 and 168 are factors)
- 168 ÷ 2 = 84 (so 2 and 84 are factors)
- 168 ÷ 3 = 56 (so 3 and 56 are factors)
- Keep checking with each number up to √168 (approx. 12.96), pairing results as you go
- List all the divisors found—these are all the factors.
This method ensures you find every factor, both small and large.
Prime Factorisation of 168
Prime factorisation breaks down 168 into its basic prime numbers. Here's how it's done:
- Divide 168 by 2: 168 ÷ 2 = 84
- Divide 84 by 2: 84 ÷ 2 = 42
- Divide 42 by 2: 42 ÷ 2 = 21
- 21 is not divisible by 2, try next prime, 3: 21 ÷ 3 = 7
- 7 is a prime number: 7 ÷ 7 = 1
So, the prime factorisation of 168 is: 2 × 2 × 2 × 3 × 7 = 168 (or \(2^3 \times 3 \times 7\)). The prime factors of 168 are 2, 3, and 7.
Worked Examples
Example 1
Find all the factors of 168 using the division method.
- Check each number from 1 upwards:
- 168 ÷ 1 = 168 ✔
- 168 ÷ 2 = 84 ✔
- 168 ÷ 3 = 56 ✔
- 168 ÷ 4 = 42 ✔
- 168 ÷ 5 = 33.6 ✘ (not a factor)
- 168 ÷ 6 = 28 ✔
- 168 ÷ 7 = 24 ✔
- 168 ÷ 8 = 21 ✔
- 168 ÷ 12 = 14 ✔
- 168 ÷ 14 = 12 ✔
- Keep going until 168 ÷ 168 = 1 ✔
So, the factors are: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.
Example 2
What number should you multiply by 21 to get 168?
- n × 21 = 168
- n = 168 ÷ 21 = 8
So, multiplying 21 by 8 gives 168.
Example 3
Find the Greatest Common Factor (GCF) of 160 and 168.
- Factors of 160: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160
- Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
- Common factors: 1, 2, 4, 8
- Greatest: 8
So, GCF (160, 168) = 8.
Practice Problems
- List all positive factors of 168.
- Write the prime factorisation of 168 in exponential form.
- How many even factors does 168 have?
- Find all the pair factors of 168.
- If 168 = m × n and m = 24, what is n?
- Find the LCM of 168 and 84 by using their prime factors.
- Which numbers less than 20 are factors of 168?
- Is 168 a perfect square? Explain why or why not.
Common Mistakes to Avoid
- Confusing factors with multiples (factors divide a number; multiples are obtained by multiplication).
- Missing factor pairs by not checking divisibility up to √168.
- Forgetting negative factors for more advanced problems.
- Omitting prime factorisation in exponent form (such as 2³ × 3 × 7).
Real-World Applications
Factors are used in everyday problems such as grouping items equally, dividing objects among people, or simplifying fractions. For example, if a baker needs to package 168 cookies into equal-sized boxes, knowing the factors helps find all the possible box sizes. Factoring also plays a role in computer security (cryptography uses large numbers’ factors), engineering, and business.
Related Topics at Vedantu
In this topic, we explored the Factors of 168, how to find them, their prime factorisation, and their significance in mathematics and real life. At Vedantu, we strive to make learning about numbers easy and clear to help you excel in exams and apply maths confidently in practical situations.
FAQs on All Factors of 168 Explained with Simple Steps
1. What are the factors of 168?
The factors of 168 are the numbers that divide 168 without leaving a remainder. These include both positive and negative values. They are: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168 (and their negative counterparts).
2. What are the prime factors of 168?
The prime factorization of 168 breaks it down into its prime components. This means expressing 168 as a product of only prime numbers. The prime factors of 168 are 2, 3, and 7. Specifically, 168 = 2³ x 3 x 7.
3. What are the pair factors of 168?
Pair factors of 168 are pairs of numbers that multiply together to equal 168. Some examples include (1, 168), (2, 84), (3, 56), (4, 42), (6, 28), (7, 24), (8, 21), (12, 14). Remember that each pair also has its negative counterparts.
4. How do I find all the factors of 168?
To find all factors, systematically divide 168 by each number starting from 1, checking for whole number results. Alternatively, use a factor tree to determine the prime factorization (2³ x 3 x 7), then systematically combine the prime factors to find all possible combinations.
5. What is the prime factorization of 168?
The prime factorization of 168 is 2³ x 3 x 7. This means 168 can be expressed as the product of its prime factors: 2 x 2 x 2 x 3 x 7.
6. How many factors does 168 have?
168 has a total of 16 positive factors. This includes 1 and 168 itself.
7. Is 168 a prime number?
No, 168 is not a prime number. A prime number has only two factors: 1 and itself. Since 168 has many factors, it's a composite number.
8. What two numbers multiply to 168?
There are several pairs of numbers that multiply to 168. Some examples include 12 x 14, 8 x 21, 6 x 28, 4 x 42. These are just a few of its many pair factors.
9. Is 168 divisible by 3?
Yes, 168 is divisible by 3. A quick way to check divisibility by 3 is to add the digits (1 + 6 + 8 = 15). Since 15 is divisible by 3, 168 is also divisible by 3.
10. How is the prime factorization of 168 useful?
The prime factorization is crucial for finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers. It simplifies calculations involving fractions and helps in various mathematical problems.
