What Are the Factors of 135?

Factors of 135 are an essential concept in number theory and arithmetic, frequently appearing in school syllabi, competitive exams, and real-life scenarios like grouping and arrangement problems. Understanding how to find the factors of 135, its prime factors, and factor pairs helps strengthen your maths skills and lays the groundwork for more advanced topics.

Understanding Factors of 135

A factor of 135 is any whole number that divides 135 exactly without leaving any remainder. In other words, if you multiply any two whole numbers and the result is 135, both numbers are factors. For example, since 3 × 45 = 135, both 3 and 45 are factors of 135. Knowing the factors is important for topics like HCF (Highest Common Factor), multiples, and simplifying fractions.

Prime Factorization of 135

To break 135 into its basic building blocks (prime numbers), we perform prime factorization:

1. Divide 135 by the smallest prime, 3: 135 ÷ 3 = 45
2. Divide 45 by 3: 45 ÷ 3 = 15
3. Divide 15 by 3: 15 ÷ 3 = 5
4. 5 is a prime number

So, the prime factorization of 135 is 3 × 3 × 3 × 5 or 3³ × 5. A factor tree visually shows how 135 splits into these primes step by step, which is often helpful in understanding for students preparing for competitive exams.

All Factors and Factor Pairs of 135

The complete list of factors of 135 can be found by dividing 135 by all whole numbers up to its square root. Whenever the division results in a whole number, both the divisor and the quotient are factors. The full list is:

• 1 × 135 = 135
• 3 × 45 = 135
• 5 × 27 = 135
• 9 × 15 = 135

Therefore, the factors of 135 are: 1, 3, 5, 9, 15, 27, 45, and 135.

How to Find Factors of 135

To find all factors of 135, start by checking each number from 1 up to 135 to see if it divides 135 exactly (i.e., no remainder). Quick tip: you only need to check up to the square root of 135 (which is about 11.6), because factors above this are just the pair partner of smaller factors. For example, if 9 divides 135 exactly (135 ÷ 9 = 15), then so does 15 (because 9 × 15 = 135).

1. Start with 1: 135 ÷ 1 = 135 (so 1 and 135 are factors)
2. Try 2: 135 ÷ 2 = 67.5 (not a whole number, so skip)
3. Try 3: 135 ÷ 3 = 45 (whole number, so 3 and 45 are factors)
4. Continue with 5, 9, 15, and 27. Each gives a whole number result, so these and their pair partners are all factors.

Any number that leaves no remainder when dividing 135 is a factor!

Prime Factorization Method

In the prime factorization method, you systematically divide 135 by its smallest possible prime numbers:

• Divide by 3: 135 ÷ 3 = 45
• Divide 45 by 3: 45 ÷ 3 = 15
• Divide 15 by 3: 15 ÷ 3 = 5
• 5 is a prime number

So, 135 = 3 × 3 × 3 × 5. This approach is also useful when finding the HCF or LCM of two numbers.

Properties and Key Observations

• 135 has 8 positive factors: 1, 3, 5, 9, 15, 27, 45, 135.
• All factors are odd (135 is an odd number).
• 135 is a composite number (more than two factors).
• Its smallest factor is 1; the greatest is 135 itself.
• Factors of 180, Factors of 105, and Factors of 12 share some similarities.

Worked Examples

Example 1: List the factors of 135

1. Start with 1 and the number itself (1 and 135).
2. Divide 135 by 3: result is 45, so 3 and 45 are factors.
3. Divide 135 by 5: result is 27, so 5 and 27 are factors.
4. Divide 135 by 9: result is 15, so 9 and 15 are factors.

All factors: 1, 3, 5, 9, 15, 27, 45, 135.

Example 2: Find the factor pairs of 135

Factor pairs are two numbers that multiply to give 135:

• (1, 135)
• (3, 45)
• (5, 27)
• (9, 15)

Example 3: Find the HCF of 135 and 45

1. Prime factorization of 135: 3 × 3 × 3 × 5
2. Prime factorization of 45: 3 × 3 × 5
3. Common prime factors: 3 × 3 × 5 = 45

Therefore, HCF of 135 and 45 is 45.

Practice Problems

• List all positive factors of 135.
• How many factor pairs does 135 have?
• What are the common factors of 135 and 180?
• Express 135 as a product of prime numbers.
• If a group of 135 students are divided equally, what is the largest possible group size (other than 135 itself)?

Common Mistakes to Avoid

• Confusing factors (divisors) with multiples (numbers that 135 makes when multiplied).
• Missing factor pairs by stopping before reaching the square root of 135.
• Incorrectly writing repeated factors (each factor should only be listed once).
• Forgetting to use prime factorization for GCF/HCF problems.

Real-World Applications

Finding factors of 135 can help in several practical uses: arranging seats, packing items into boxes, or dividing students into equal groups. For example, if you have 135 apples and want to pack them equally, knowing the factors tells you the ways to group them without leftovers. Factorization is also used in topics such as cryptography and optimizing resources.

In summary, learning about the factors of 135 and their applications is crucial for excelling in maths exams and solving practical problems. At Vedantu, we simplify such maths concepts, offering clear explanations, examples, and practice to help you boost confidence and master number theory fundamentals. For related topics, explore Factors of a Number and Prime Numbers on Vedantu.