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GCF of 34 and 38

GCF of 34 and 38 is the largest possible number that divides 34 and 38 exactly without any remainder. The factors of 34 and 38 are 1, 2, 17, 34 and 1, 2, 19, 38 respectively. There are 3 commonly used methods to find the GCF of 34 and 38 - long division, prime factorization, and Euclidean algorithm.

1.	GCF of 34 and 38
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 34 and 38?

Answer: GCF of 34 and 38 is 2.

Explanation:

The GCF of two non-zero integers, x(34) and y(38), is the greatest positive integer m(2) that divides both x(34) and y(38) without any remainder.

Methods to Find GCF of 34 and 38

Let's look at the different methods for finding the GCF of 34 and 38.

Prime Factorization Method
Listing Common Factors
Using Euclid's Algorithm

GCF of 34 and 38 by Prime Factorization

Prime factorization of 34 and 38 is (2 × 17) and (2 × 19) respectively. As visible, 34 and 38 have only one common prime factor i.e. 2. Hence, the GCF of 34 and 38 is 2.

GCF of 34 and 38 by Listing Common Factors

Factors of 34: 1, 2, 17, 34
Factors of 38: 1, 2, 19, 38

There are 2 common factors of 34 and 38, that are 1 and 2. Therefore, the greatest common factor of 34 and 38 is 2.

GCF of 34 and 38 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 = 38 and Y = 34

GCF(38, 34) = GCF(34, 38 mod 34) = GCF(34, 4)
GCF(34, 4) = GCF(4, 34 mod 4) = GCF(4, 2)
GCF(4, 2) = GCF(2, 4 mod 2) = GCF(2, 0)
GCF(2, 0) = 2 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 34 and 38 is 2.

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GCF of 34 and 38 Examples

Example 1: Find the greatest number that divides 34 and 38 exactly.

Solution:

The greatest number that divides 34 and 38 exactly is their greatest common factor, i.e. GCF of 34 and 38.
⇒ Factors of 34 and 38:
- Factors of 34 = 1, 2, 17, 34
- Factors of 38 = 1, 2, 19, 38
Therefore, the GCF of 34 and 38 is 2.
Example 2: For two numbers, GCF = 2 and LCM = 646. If one number is 38, find the other number.

Solution:

Given: GCF (x, 38) = 2 and LCM (x, 38) = 646
∵ GCF × LCM = 38 × (x)
⇒ x = (GCF × LCM)/38
⇒ x = (2 × 646)/38
⇒ x = 34
Therefore, the other number is 34.
Example 3: Find the GCF of 34 and 38, if their LCM is 646.

Solution:

∵ LCM × GCF = 34 × 38
⇒ GCF(34, 38) = (34 × 38)/646 = 2
Therefore, the greatest common factor of 34 and 38 is 2.

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FAQs on GCF of 34 and 38

What is the GCF of 34 and 38?

The GCF of 34 and 38 is 2. To calculate the GCF of 34 and 38, we need to factor each number (factors of 34 = 1, 2, 17, 34; factors of 38 = 1, 2, 19, 38) and choose the greatest factor that exactly divides both 34 and 38, i.e., 2.

What are the Methods to Find GCF of 34 and 38?

There are three commonly used methods to find the GCF of 34 and 38.

By Long Division
By Prime Factorization
By Listing Common Factors

What is the Relation Between LCM and GCF of 34, 38?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 34 and 38, i.e. GCF × LCM = 34 × 38.

If the GCF of 38 and 34 is 2, Find its LCM.

GCF(38, 34) × LCM(38, 34) = 38 × 34
Since the GCF of 38 and 34 = 2
⇒ 2 × LCM(38, 34) = 1292
Therefore, LCM = 646
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How to Find the GCF of 34 and 38 by Long Division Method?

To find the GCF of 34, 38 using long division method, 38 is divided by 34. The corresponding divisor (2) when remainder equals 0 is taken as GCF.

How to Find the GCF of 34 and 38 by Prime Factorization?

To find the GCF of 34 and 38, we will find the prime factorization of the given numbers, i.e. 34 = 2 × 17; 38 = 2 × 19.
⇒ Since 2 is the only common prime factor of 34 and 38. Hence, GCF (34, 38) = 2.
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