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GCF of 34 and 51

GCF of 34 and 51 is the largest possible number that divides 34 and 51 exactly without any remainder. The factors of 34 and 51 are 1, 2, 17, 34 and 1, 3, 17, 51 respectively. There are 3 commonly used methods to find the GCF of 34 and 51 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 34 and 51
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 34 and 51?

Answer: GCF of 34 and 51 is 17.

Explanation:

The GCF of two non-zero integers, x(34) and y(51), is the greatest positive integer m(17) that divides both x(34) and y(51) without any remainder.

Methods to Find GCF of 34 and 51

The methods to find the GCF of 34 and 51 are explained below.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

GCF of 34 and 51 by Euclidean Algorithm

As per the Euclidean Algorithm, GCF(X, Y) = GCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

Here X = 51 and Y = 34

GCF(51, 34) = GCF(34, 51 mod 34) = GCF(34, 17)
GCF(34, 17) = GCF(17, 34 mod 17) = GCF(17, 0)
GCF(17, 0) = 17 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 34 and 51 is 17.

GCF of 34 and 51 by Prime Factorization

Prime factorization of 34 and 51 is (2 × 17) and (3 × 17) respectively. As visible, 34 and 51 have only one common prime factor i.e. 17. Hence, the GCF of 34 and 51 is 17.

GCF of 34 and 51 by Long Division

GCF of 34 and 51 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 51 (larger number) by 34 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (34) by the remainder (17).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (17) is the GCF of 34 and 51.

☛ Also Check:

GCF of 34 and 51 Examples

Example 1: Find the GCF of 34 and 51, if their LCM is 102.

Solution:

∵ LCM × GCF = 34 × 51
⇒ GCF(34, 51) = (34 × 51)/102 = 17
Therefore, the greatest common factor of 34 and 51 is 17.
Example 2: For two numbers, GCF = 17 and LCM = 102. If one number is 34, find the other number.

Solution:

Given: GCF (x, 34) = 17 and LCM (x, 34) = 102
∵ GCF × LCM = 34 × (x)
⇒ x = (GCF × LCM)/34
⇒ x = (17 × 102)/34
⇒ x = 51
Therefore, the other number is 51.
Example 3: Find the greatest number that divides 34 and 51 exactly.

Solution:

The greatest number that divides 34 and 51 exactly is their greatest common factor, i.e. GCF of 34 and 51.
⇒ Factors of 34 and 51:
- Factors of 34 = 1, 2, 17, 34
- Factors of 51 = 1, 3, 17, 51
Therefore, the GCF of 34 and 51 is 17.

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FAQs on GCF of 34 and 51

What is the GCF of 34 and 51?

The GCF of 34 and 51 is 17. To calculate the greatest common factor of 34 and 51, we need to factor each number (factors of 34 = 1, 2, 17, 34; factors of 51 = 1, 3, 17, 51) and choose the greatest factor that exactly divides both 34 and 51, i.e., 17.

What is the Relation Between LCM and GCF of 34, 51?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 34 and 51, i.e. GCF × LCM = 34 × 51.

How to Find the GCF of 34 and 51 by Prime Factorization?

To find the GCF of 34 and 51, we will find the prime factorization of the given numbers, i.e. 34 = 2 × 17; 51 = 3 × 17.
⇒ Since 17 is the only common prime factor of 34 and 51. Hence, GCF (34, 51) = 17.
☛ Prime Numbers

How to Find the GCF of 34 and 51 by Long Division Method?

To find the GCF of 34, 51 using long division method, 51 is divided by 34. The corresponding divisor (17) when remainder equals 0 is taken as GCF.

If the GCF of 51 and 34 is 17, Find its LCM.

GCF(51, 34) × LCM(51, 34) = 51 × 34
Since the GCF of 51 and 34 = 17
⇒ 17 × LCM(51, 34) = 1734
Therefore, LCM = 102
☛ GCF Calculator

What are the Methods to Find GCF of 34 and 51?

There are three commonly used methods to find the GCF of 34 and 51.

By Listing Common Factors
By Long Division
By Prime Factorization

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