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106 LearnersLast updated on September 24, 2025
GCF of 12 and 75
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 12 and 75.
What is the GCF of 12 and 75?
The greatest common factor of 12 and 75 is 3. 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 12 and 75?
To find the GCF of 12 and 75, a few methods are described below -
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 12 and 75 by Using Listing of factors
Steps to find the GCF of 12 and 75 using the listing of factors
Step 1: Firstly, list the factors of each number
Factors of 12 = 1, 2, 3, 4, 6, 12.
Factors of 75 = 1, 3, 5, 15, 25, 75.
Step 2: Now, identify the common factors of them Common factors of 12 and 75: 1, 3.
Step 3: Choose the largest factor The largest factor that both numbers have is 3.
The GCF of 12 and 75 is 3.
GCF of 12 and 75 Using Prime Factorization
To find the GCF of 12 and 75 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 12: 12 = 2 x 2 x 3 = 2² x 3
Prime Factors of 75: 75 = 3 x 5 x 5 = 3 x 5²
Step 2: Now, identify the common prime factors The common prime factor is: 3
Step 3: Multiply the common prime factors The Greatest Common Factor of 12 and 75 is 3.
GCF of 12 and 75 Using Division Method or Euclidean Algorithm Method
Find the GCF of 12 and 75 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number
Here, divide 75 by 12 75 ÷ 12 = 6 (quotient), The remainder is calculated as 75 − (12×6) = 3 The remainder is 3, not zero, so continue the process
Step 2: Now divide the previous divisor (12) by the previous remainder (3) Divide 12 by 3 12 ÷ 3 = 4 (quotient), remainder = 12 − (3×4) = 0
The remainder is zero, the divisor will become the GCF. The GCF of 12 and 75 is 3.
Common Mistakes and How to Avoid Them in GCF of 12 and 75
Finding GCF of 12 and 75 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 75, 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 that 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 12 and 75 Examples
Problem 1
A gardener has 12 rose bushes and 75 tulip plants. She wants to arrange them into the largest possible equal groups. How many plants will be in each group?
We should find the GCF of 12 and 75 GCF of 12 and 75 is 3.
There will be 3 equal groups. 12 ÷ 3 = 4 75 ÷ 3 = 25
There will be 3 groups, and each group gets 4 rose bushes and 25 tulip plants.
Explanation
As the GCF of 12 and 75 is 3, the gardener can make 3 groups.
Now divide 12 and 75 by 3.
Each group gets 4 rose bushes and 25 tulip plants.
Problem 2
A chef has 12 kilograms of flour and 75 kilograms of sugar. He wants to make batches of the largest equal size possible. How many kilograms of each ingredient will be in each batch?
GCF of 12 and 75 is 3.
So each batch will have 3 kilograms of each ingredient.
Explanation
There are 12 kilograms of flour and 75 kilograms of sugar.
To find the total amount in each batch, we should find the GCF of 12 and 75.
There will be 3 kilograms of each ingredient in each batch.
Problem 3
A library has 12 fiction books and 75 non-fiction books. The librarian wants to organize them in sections with the same number of books, using the largest possible number of books per section. How many books will be in each section?
For calculating the largest equal number of books, we have to calculate the GCF of 12 and 75.
The GCF of 12 and 75 is 3. The sections will have 3 books each.
Explanation
For calculating the largest number of books per section, first, we need to calculate the GCF of 12 and 75, which is 3.
The number of books in each section will be 3.
Problem 4
A workshop has two lengths of wire, one 12 meters long and the other 75 meters long. The technician wants to cut them into the longest possible equal pieces, without any wire left over. What should be the length of each piece?
The technician needs the longest piece of wire GCF of 12 and 75 is 3.
The longest length of each piece is 3 meters.
Explanation
To find the longest length of each piece of wire, 12 meters and 75 meters, respectively,
we have to find the GCF of 12 and 75, which is 3 meters.
The longest length of each piece is 3 meters.
Problem 5
If the GCF of 12 and ‘b’ is 3, and the LCM is 300. Find ‘b’.
The value of ‘b’ is 75.
Explanation
GCF x LCM = product of the numbers
3 × 300 = 12 × b
900 = 12b
b = 900 ÷ 12 = 75
FAQs on the Greatest Common Factor of 12 and 75
1.What is the LCM of 12 and 75?
2.Is 75 divisible by 5?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 75?
5.Are 12 and 75 prime numbers?
Important Glossaries for GCF of 12 and 75
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- 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 12 are 2 and 3.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 75 is divided by 12, the remainder is 3.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 12 and 75 is 300.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 12 and 75 is 3, as it is their largest common factor that divides the numbers completely.
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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.
