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101 LearnersLast updated on September 11, 2025
GCF of 51 and 85
The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share the items equally, to group or arrange items, and schedule events. In this topic, we will learn about the GCF of 51 and 85.
What is the GCF of 51 and 85?
The greatest common factor of 51 and 85 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 51 and 85?
To find the GCF of 51 and 85, a few methods are described below:
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 51 and 85 by Using Listing of Factors
Steps to find the GCF of 51 and 85 using the listing of factors:
Step 1: Firstly, list the factors of each number. Factors of 51 = 1, 3, 17, 51. Factors of 85 = 1, 5, 17, 85.
Step 2: Now, identify the common factors of them. Common factors of 51 and 85: 1, 17.
Step 3: Choose the largest factor. The largest factor that both numbers have is 17. The GCF of 51 and 85 is 17.
GCF of 51 and 85 Using Prime Factorization
To find the GCF of 51 and 85 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number.
Prime Factors of 51: 51 = 3 x 17.
Prime Factors of 85: 85 = 5 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 51 and 85 is 17.
GCF of 51 and 85 Using Division Method or Euclidean Algorithm Method
Find the GCF of 51 and 85 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number. Here, divide 85 by 51. 85 ÷ 51 = 1 (quotient), The remainder is calculated as 85 − (51×1) = 34. The remainder is 34, not zero, so continue the process.
Step 2: Now divide the previous divisor (51) by the previous remainder (34). 51 ÷ 34 = 1 (quotient), remainder = 51 − (34×1) = 17.
Step 3: Now divide the previous divisor (34) by the previous remainder (17). 34 ÷ 17 = 2 (quotient), remainder = 34 − (17×2) = 0.
The remainder is zero, the divisor will become the GCF. The GCF of 51 and 85 is 17.
Common Mistakes and How to Avoid Them in GCF of 51 and 85
Finding the GCF of 51 and 85 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 51, students may mention 2, 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 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 51 and 85 Examples
Problem 1
A gardener has 51 roses and 85 tulips. She wants to group them into equal bouquets, with the largest number of flowers in each bouquet. How many flowers will be in each bouquet?
We should find the GCF of 51 and 85.
GCF of 51 and 85 is 17. 51 ÷ 17 = 3
85 ÷ 17 = 5
There will be 17 bouquets, and each bouquet gets 3 roses and 5 tulips.
Explanation
As the GCF of 51 and 85 is 17, the gardener can make 17 bouquets. Now divide 51 and 85 by 17. Each bouquet gets 3 roses and 5 tulips.
Problem 2
A chef has 51 apples and 85 oranges. He wants to arrange them in trays with the same number of fruits in each tray, using the largest possible number of fruits per tray. How many fruits will be in each tray?
GCF of 51 and 85 is 17. So each tray will have 17 fruits.
Explanation
There are 51 apples and 85 oranges. To find the total number of fruits in each tray, we should find the GCF of 51 and 85. There will be 17 fruits in each tray.
Problem 3
A painter has 51 meters of red tape and 85 meters of blue tape. She wants to cut both tapes into pieces of equal length, using the longest possible length. What should be the length of each piece?
To calculate the longest equal length, we have to calculate the GCF of 51 and 85.
The GCF of 51 and 85 is 17. The tape is 17 meters long.
Explanation
To calculate the longest length of the tape, first, we need to calculate the GCF of 51 and 85, which is 17. The length of each piece of the tape will be 17 meters.
Problem 4
A musician has two strings, one 51 cm long and the other 85 cm long. He wants to cut them into the longest possible equal pieces, without any string left over. What should be the length of each piece?
The musician needs the longest piece of string. GCF of 51 and 85 is 17.
The longest length of each piece is 17 cm.
Explanation
To find the longest length of each piece of the two strings, 51 cm and 85 cm, respectively, we have to find the GCF of 51 and 85, which is 17 cm. The longest length of each piece is 17 cm.
Problem 5
If the GCF of 51 and ‘b’ is 17, and the LCM is 255. Find ‘b’.
The value of ‘b’ is 85.
Explanation
GCF x LCM = product of the numbers
17 × 255 = 51 × b
4335 = 51b
b = 4335 ÷ 51 = 85
FAQs on the Greatest Common Factor of 51 and 85
1.What is the LCM of 51 and 85?
2.Is 51 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 85?
5.Are 51 and 85 prime numbers?
Important Glossaries for GCF of 51 and 85
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 17 are 1 and 17.
- 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, 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 51 are 3 and 17.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 85 is divided by 51, the remainder is 34, and the quotient is 1.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 51 and 85 is 255.
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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.
