Factors of 168 Complete Guide with Prime Factorization

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:

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:

1. Start with 1 and move upwards: 168 ÷ 1 = 168 (so 1 and 168 are factors)
2. 168 ÷ 2 = 84 (so 2 and 84 are factors)
3. 168 ÷ 3 = 56 (so 3 and 56 are factors)
4. Keep checking with each number up to √168 (approx. 12.96), pairing results as you go
5. 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:

1. Divide 168 by 2: 168 ÷ 2 = 84
2. Divide 84 by 2: 84 ÷ 2 = 42
3. Divide 42 by 2: 42 ÷ 2 = 21
4. 21 is not divisible by 2, try next prime, 3: 21 ÷ 3 = 7
5. 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.

1. Check each number from 1 upwards:
2. 168 ÷ 1 = 168 ✔
3. 168 ÷ 2 = 84 ✔
4. 168 ÷ 3 = 56 ✔
5. 168 ÷ 4 = 42 ✔
6. 168 ÷ 5 = 33.6 ✘ (not a factor)
7. 168 ÷ 6 = 28 ✔
8. 168 ÷ 7 = 24 ✔
9. 168 ÷ 8 = 21 ✔
10. 168 ÷ 12 = 14 ✔
11. 168 ÷ 14 = 12 ✔
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?

1. n × 21 = 168
2. n = 168 ÷ 21 = 8

So, multiplying 21 by 8 gives 168.

Example 3

Find the Greatest Common Factor (GCF) of 160 and 168.

1. Factors of 160: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160
2. Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
3. Common factors: 1, 2, 4, 8
4. 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

• Factors of 120
• Factors of 84
• Prime Numbers
• Prime Factors Explained
• Common Factors

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.