107 LearnersLast updated on August 6, 2025
GCF of 34 and 51
The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share items equally, group or arrange items, and schedule events. In this topic, we will learn about the GCF of 34 and 51.
What is the GCF of 34 and 51?
The greatest common factor of 34 and 51 is 17. The largest divisor of two or more numbers is called the GCF of the numbers. 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 34 and 51?
To find the GCF of 34 and 51, a few methods are described below:
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 34 and 51 by Using Listing of Factors
Steps to find the GCF of 34 and 51 using the listing of factors:
Step 1: Firstly, list the factors of each number:
Factors of 34 = 1, 2, 17, 34.
Factors of 51 = 1, 3, 17, 51.
Step 2: Now, identify the common factors of them.
Common factors of 34 and 51: 1, 17.
Step 3: Choose the largest factor:
The largest factor that both numbers have is 17.
The GCF of 34 and 51 is 17.
GCF of 34 and 51 Using Prime Factorization
To find the GCF of 34 and 51 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number:
Prime Factors of 34: 34 = 2 x 17
Prime Factors of 51: 51 = 3 x 17
Step 2: Now, identify the common prime factors:
The common prime factor is: 17
Step 3: Multiply the common prime factors.
The Greatest Common Factor of 34 and 51 is 17.
GCF of 34 and 51 Using Division Method or Euclidean Algorithm Method
Find the GCF of 34 and 51 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number.
Here, divide 51 by 34 51 ÷ 34 = 1 (quotient), The remainder is calculated as 51 − (34x1) = 17
The remainder is 17, not zero, so continue the process
Step 2: Now divide the previous divisor (34) by the previous remainder (17)
Divide 34 by 17 34 ÷ 17 = 2 (quotient), remainder = 34 − (17x2) = 0
The remainder is zero, the divisor will become the GCF.
The GCF of 34 and 51 is 17.
Common Mistakes and How to Avoid Them in GCF of 34 and 51
Finding GCF of 34 and 51 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 34, 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 indicates an incomplete understanding of factors. Students should include 1 as a factor.
Mistake 4
Using Multiples Instead of Factors
Students confuse factors with 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 34 and 51 Examples
Problem 1
A gardener has 34 tulips and 51 daisies. She wants to plant them in equal rows with the largest number of flowers in each row. How many flowers will be in each row?
We should find the GCF of 34 and 51.
GCF of 34 and 51 is 17.
There are 17 equal groups.
34 ÷ 17 = 2
51 ÷ 17 = 3
There will be 17 groups, and each row gets 2 tulips and 3 daisies.
Explanation
As the GCF of 34 and 51 is 17, the gardener can make 17 groups.
Now divide 34 and 51 by 17.
Each row gets 2 tulips and 3 daisies.
Problem 2
A bakery has 34 croissants and 51 muffins. They want to package them in boxes with the same number of items in each box, using the largest possible number of items per box. How many items will be in each box?
GCF of 34 and 51 is 17.
So each box will have 17 items.
Explanation
There are 34 croissants and 51 muffins.
To find the total number of items in each box, we should find the GCF of 34 and 51.
There will be 17 items in each box.
Problem 3
A tailor has 34 meters of silk and 51 meters of cotton 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 34 and 51.
The GCF of 34 and 51 is 17.
The fabric is 17 meters long.
Explanation
For calculating the longest length of the fabric, first we need to calculate the GCF of 34 and 51, which is 17.
The length of each piece of fabric will be 17 meters.
Problem 4
A carpenter has two wooden planks, one 34 cm long and the other 51 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 34 and 51 is 17.
The longest length of each piece is 17 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 34 cm and 51 cm, respectively, we have to find the GCF of 34 and 51, which is 17 cm. The longest length of each piece is 17 cm.
Problem 5
If the GCF of 34 and ‘b’ is 17, and the LCM is 102. Find ‘b’.
The value of ‘b’ is 51.
Explanation
GCF x LCM = product of the numbers
17 x 102 = 34 x b
1734 = 34b
b = 1734 ÷ 34 = 51
FAQs on the Greatest Common Factor of 34 and 51
1.What is the LCM of 34 and 51?
2.Is 34 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 51?
5.Are 34 and 51 prime numbers?
6.How can children in United States use numbers in everyday life to understand GCF of 34 and 51?
7.What are some fun ways kids in United States can practice GCF of 34 and 51 with numbers?
8.What role do numbers and GCF of 34 and 51 play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve GCF of 34 and 51 skills?
Important Glossaries for GCF of 34 and 51
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 17 are 1 and 17.
- 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 51 are 3 and 17.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 51 is divided by 34, the remainder is 17.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 34 and 51 is 102.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 34 and 51 is 17, as it is their largest common factor that divides the numbers completely.
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.
