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106 LearnersLast updated on September 24, 2025
GCF of 135 and 180
The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share items equally, to group or arrange items, and schedule events. In this topic, we will learn about the GCF of 135 and 180.
What is the GCF of 135 and 180?
The greatest common factor of 135 and 180 is 45. The largest divisor of two or more numbers is called the GCF of the number.
If two numbers are co-prime, they have no common factors other than 1, so their GCF is 1.
The GCF of two numbers cannot be negative because divisors are always positive.
How to find the GCF of 135 and 180?
To find the GCF of 135 and 180, a few methods are described below -
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 135 and 180 by Using Listing of Factors
Steps to find the GCF of 135 and 180 using the listing of factors
Step 1: Firstly, list the factors of each number
Factors of 135 = 1, 3, 5, 9, 15, 27, 45, 135.
Factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
Step 2: Now, identify the common factors of them Common factors of 135 and 180: 1, 3, 5, 9, 15, 45.
Step 3: Choose the largest factor The largest factor that both numbers have is 45.
The GCF of 135 and 180 is 45.
GCF of 135 and 180 Using Prime Factorization
To find the GCF of 135 and 180 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 135: 135 = 3 × 3 × 3 × 5 = 3³ × 5
Prime Factors of 180: 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
Step 2: Now, identify the common prime factors The common prime factors are: 3² × 5
Step 3: Multiply the common prime factors 3² × 5 = 9 × 5 = 45.
The Greatest Common Factor of 135 and 180 is 45.
GCF of 135 and 180 Using Division Method or Euclidean Algorithm Method
Find the GCF of 135 and 180 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 180 by 135 180 ÷ 135 = 1 (quotient), The remainder is calculated as 180 − (135×1) = 45
The remainder is 45, not zero, so continue the process
Step 2: Now divide the previous divisor (135) by the previous remainder (45) Divide 135 by 45 135 ÷ 45 = 3 (quotient), remainder = 135 − (45×3) = 0
The remainder is zero, the divisor will become the GCF. The GCF of 135 and 180 is 45.
Common Mistakes and How to Avoid Them in GCF of 135 and 180
Finding the GCF of 135 and 180 looks simple, but students often make mistakes while calculating the GCF. Here are some common mistakes to be avoided by the students.
Mistake 1
Listing Incorrect Factors
Students may sometimes list incorrect factors. For example, while listing factors of 135, students may mention 10, which is incorrect. To avoid this, students should carefully divide the number and list the factors correctly.
Mistake 2
Choosing the Wrong Common Factor
Students may sometimes select the smallest common factor instead of the largest one. To avoid this confusion, students should list all the common factors and find the greatest one.
Mistake 3
Forgetting to Include 1 as a Factor
Sometimes students may forget 1 as a common factor of the numbers. However, it does not affect the GCF, but it tells about the incomplete understanding of the factors. Students should include 1 as a factor.
Mistake 4
Using Multiples Instead of Factors
Students confuse factors and multiples. In that confusion, sometimes they may write multiples instead of factors. To avoid this confusion, students should know the definitions of multiples and factors clearly.
Mistake 5
Assuming GCF is Always an Even Number
Students may assume that the GCF of two numbers will always be an even number. But it's not true; a GCF can also be an odd number. To avoid this, students should focus on common factors rather than focusing on even and odd numbers.
Greatest Common Factor of 135 and 180 Examples
Problem 1
A gardener has 135 tulips and 180 roses. She wants to plant them in equal groups, with the largest number of flowers in each group. How many flowers will be in each group?
We should find the GCF of 135 and 180 GCF of 135 and 180 3² × 5 = 9 × 5 = 45.
There are 45 equal groups 135 ÷ 45 = 3 180 ÷ 45 = 4
There will be 45 groups, and each group gets 3 tulips and 4 roses.
Explanation
As the GCF of 135 and 180 is 45, the gardener can make 45 groups.
Now divide 135 and 180 by 45. Each group gets 3 tulips and 4 roses.
Problem 2
A school library has 135 fiction books and 180 non-fiction books. They want to arrange them on shelves with the same number of books on each shelf, using the largest possible number of books per shelf. How many books will be on each shelf?
GCF of 135 and 180 3² × 5 = 9 × 5 = 45.
So each shelf will have 45 books.
Explanation
There are 135 fiction and 180 non-fiction books.
To find the total number of books on each shelf, we should find the GCF of 135 and 180.
There will be 45 books on each shelf.
Problem 3
A tailor has 135 meters of red fabric and 180 meters of blue fabric. She wants to cut both fabrics into pieces of equal length, using the longest possible length. What should be the length of each piece?
For calculating the longest equal length,
we have to calculate the GCF of 135 and 180 The GCF of 135 and 180 3² × 5 = 9 × 5 = 45.
The fabric is 45 meters long.
Explanation
For calculating the longest length of the fabric, first, we need to calculate the GCF of 135 and 180, which is 45.
The length of each piece of the fabric will be 45 meters.
Problem 4
A carpenter has two wooden boards, one 135 cm long and the other 180 cm long. He wants to cut them into the longest possible equal pieces, without any wood left over. What should be the length of each piece?
The carpenter needs the longest piece of wood GCF of 135 and 180 3² × 5 = 9 × 5 = 45.
The longest length of each piece is 45 cm.
Explanation
To find the longest length of each piece of the two wooden boards, 135 cm and 180 cm, respectively,
we have to find the GCF of 135 and 180, which is 45 cm.
The longest length of each piece is 45 cm.
Problem 5
If the GCF of 135 and ‘b’ is 45, and the LCM is 540, find ‘b’.
The value of ‘b’ is 180.
Explanation
GCF × LCM = product of the numbers
45 × 540 = 135 × b
24,300 = 135b
b = 24,300 ÷ 135 = 180
FAQs on the Greatest Common Factor of 135 and 180
1.What is the LCM of 135 and 180?
2.Is 135 divisible by 5?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 180?
5.Are 135 and 180 prime numbers?
Important Glossaries for GCF of 135 and 180
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 45 are 1, 3, 5, 9, 15, and 45.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 5 are 5, 10, 15, 20, 25, and so on.
- Prime Factors: These are the factors of a number that are prime numbers and divide the given number completely. For example, the prime factors of 180 are 2, 3, and 5.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 13 is divided by 4, the remainder is 1 and the quotient is 3.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 135 and 180 is 540.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
