The factors of 135 are the numbers that divide 135 completely without leaving any remainder. In other words, the numbers which are multiplied in pairs resulting in the number 135 are the factors of 135. The factors can be expressed in both positive and negative form, but the factors cannot be in decimal or fractional form. For example, the pair factor of 135 is represented by (1, 135) or (-1, -135). Therefore, the multiplication of negative numbers, such as multiplying -1 and -135, will result in the original number 135. In this article, we will learn the factors of 135, positive and negative pair factors of 135, how to find the prime factors of 135 using the prime factorization method, and many solved examples.
Table of Contents:
- What are the Factors of 135?
- Pair Factors of 135
- Factors of 135 by Division Method
- Prime Factorization of 135
- Solved Examples
- FAQs
What are the Factors of 135?
The numbers that divide the given number evenly without leaving any remainder are called the factors of a number. Thus, the factors of 135 are the numbers that divide 135 completely and leave a remainder 0. As 135 is a composite number, it has many factors other than 1 and 135. Hence, the factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135. Similarly, the negative factors of 135 are -1, -3, -5, -9, -15, -27, -45 and -135.
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Factors of 135: 1, 3, 5, 9, 15, 27, 45 and 135. Prime Factorization of 135: 3 × 3 × 3 × 5 or 33 × 5 |
Pair Factors of 135
The numbers that are multiplied together in pairs resulting in the original number 135 are the pair factors of 135. As discussed above, the pair factors of 135 can be expressed in both positive and negative form. The following are the positive and negative pair factors of 135.
Positive Pair Factors of 135:
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Positive Factors of 135 |
Positive Pair Factors of 135 |
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1 × 135 |
(1, 135) |
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3 × 45 |
(3, 45) |
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5 × 27 |
(5, 27) |
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9 × 15 |
(9, 15) |
Therefore, the positive pair factors of 135 are (1, 135), (3, 45), (5, 27), and (9, 15).
Negative Pair Factors of 135:
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Negative Factors of 135 |
Negative Pair Factors of 135 |
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-1 × -135 |
(-1, -135) |
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-3 × -45 |
(-3, -45) |
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-5 × -27 |
(-5, -27) |
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-9 × -15 |
(-9, -15) |
Therefore, the negative pair factors of 135 are (-1, -135), (-3, -45), (-5, -27), and (-9, -15).
Factors of 135 by Division Method
In the division method, the factors of 135 are found by dividing 135 by different integers. If the integers divide 135 evenly without leaving any remainder value, then those integers are the factors of 135. Now, let us discuss how to find the factors of 135 using the division method.
- 135/1 = 135 (Factor is 1 and Remainder is 0)
- 135/3 = 45 (Factor is 3 and Remainder is 0)
- 135/5 = 27 (Factor is 5 and Remainder is 0)
- 135/9 = 15 (Factor is 9 and Remainder is 0)
- 135/15 = 9 (Factor is 15 and Remainder is 0)
- 135/27 = 5 (Factor is 27 and Remainder is 0)
- 135/45 = 3 (Factor is 45 and Remainder is 0)
- 135/135 = 1 (Factor is 135 and Remainder is 0)
If we divide 135 by any numbers other than 1, 3, 5, 9, 15, 27, 45 and 135, it leaves a remainder value. Hence, the factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135.
Prime Factorization of 135
The process of writing the number 135 as the product of its prime factors is called the prime factorization of 135. Go through the following procedure to find the prime factors of 135.
Take a pair factor of 135, say (1, 135)
Here, 135 is a composite number, which can be further split as follows:
135 = 9 × 15
Now, both 9 and 15 are composite numbers that can split again as:
9 = 3 × 3
15 = 3 × 5
Where 3 and 5 are prime numbers.
Now, write all the numbers in the form of the product of its prime factors.
Thus, 135 is written as 3 × 3 × 3 × 5
Therefore, the prime factorization of 135 is 3 × 3 × 3 × 5 or 33 × 5.
Video Lesson on Prime Factors
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Solved Examples
Example 1:
Find the common factors of 135 and 134.
Solution:
The factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135.
The factors of 134 are 1, 2, 67 and 134.
Thus, the common factor of 135 and 134 is 1.
Example 2:
What are the common factors of 135 and 136?
Solution:
Factors of 135 = 1, 3, 5, 9, 15, 27, 45 and 135.
Factors of 136 = 1, 2, 4 , 8, 17, 34, 68 and 136.
Hence, the common factor of 135 and 134 is 1.
Example 3:
Find the common factors of 135 and 137.
Solution:
The factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135.
The factors of 137 are 1 and 137.
As 137 is a prime number, the common factor of 135 and 137 is 1 only.
Stay tuned with BYJU’S – The Learning App to learn more about different factors of a number and other interesting Maths concepts.
Frequently Asked Questions on Factors of 135
What are the factors of 135?
The factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135.
What is the prime factorization of 135?
The prime factorization of 135 is 3 × 3 × 3 × 5 or 33 × 5.
What are the positive pair factors of 135?
The positive pair factors of 135 are (1, 135), (3, 45), (5, 27), and (9, 15).
What are the negative pair factors of 135?
The negative pair factors of 135 are (-1, -135), (-3, -45), (-5, -27), and (-9, -15).
Is 45 a factor of 135?
Yes, 45 is a factor of 135. If 135 is divisible by 45, it leaves a quotient value of 3 and the remainder value as 0. Hence, 45 is a factor of 135.
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