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GCF of 68 and 102

GCF of 68 and 102 is the largest possible number that divides 68 and 102 exactly without any remainder. The factors of 68 and 102 are 1, 2, 4, 17, 34, 68 and 1, 2, 3, 6, 17, 34, 51, 102 respectively. There are 3 commonly used methods to find the GCF of 68 and 102 - Euclidean algorithm, long division, and prime factorization.

1.	GCF of 68 and 102
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 68 and 102?

Answer: GCF of 68 and 102 is 34.

Explanation:

The GCF of two non-zero integers, x(68) and y(102), is the greatest positive integer m(34) that divides both x(68) and y(102) without any remainder.

Methods to Find GCF of 68 and 102

The methods to find the GCF of 68 and 102 are explained below.

Prime Factorization Method
Using Euclid's Algorithm
Listing Common Factors

GCF of 68 and 102 by Prime Factorization

Prime factorization of 68 and 102 is (2 × 2 × 17) and (2 × 3 × 17) respectively. As visible, 68 and 102 have common prime factors. Hence, the GCF of 68 and 102 is 2 × 17 = 34.

GCF of 68 and 102 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 = 102 and Y = 68

GCF(102, 68) = GCF(68, 102 mod 68) = GCF(68, 34)
GCF(68, 34) = GCF(34, 68 mod 34) = GCF(34, 0)
GCF(34, 0) = 34 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 68 and 102 is 34.

GCF of 68 and 102 by Listing Common Factors

Factors of 68: 1, 2, 4, 17, 34, 68
Factors of 102: 1, 2, 3, 6, 17, 34, 51, 102

There are 4 common factors of 68 and 102, that are 1, 2, 34, and 17. Therefore, the greatest common factor of 68 and 102 is 34.

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GCF of 68 and 102 Examples

Example 1: The product of two numbers is 6936. If their GCF is 34, what is their LCM?

Solution:

Given: GCF = 34 and product of numbers = 6936
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 6936/34
Therefore, the LCM is 204.
Example 2: Find the greatest number that divides 68 and 102 exactly.

Solution:

The greatest number that divides 68 and 102 exactly is their greatest common factor, i.e. GCF of 68 and 102.
⇒ Factors of 68 and 102:
- Factors of 68 = 1, 2, 4, 17, 34, 68
- Factors of 102 = 1, 2, 3, 6, 17, 34, 51, 102
Therefore, the GCF of 68 and 102 is 34.
Example 3: Find the GCF of 68 and 102, if their LCM is 204.

Solution:

∵ LCM × GCF = 68 × 102
⇒ GCF(68, 102) = (68 × 102)/204 = 34
Therefore, the greatest common factor of 68 and 102 is 34.

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FAQs on GCF of 68 and 102

What is the GCF of 68 and 102?

The GCF of 68 and 102 is 34. To calculate the greatest common factor (GCF) of 68 and 102, we need to factor each number (factors of 68 = 1, 2, 4, 17, 34, 68; factors of 102 = 1, 2, 3, 6, 17, 34, 51, 102) and choose the greatest factor that exactly divides both 68 and 102, i.e., 34.

What is the Relation Between LCM and GCF of 68, 102?

The following equation can be used to express the relation between LCM and GCF of 68 and 102, i.e. GCF × LCM = 68 × 102.

What are the Methods to Find GCF of 68 and 102?

There are three commonly used methods to find the GCF of 68 and 102.

By Euclidean Algorithm
By Long Division
By Prime Factorization

If the GCF of 102 and 68 is 34, Find its LCM.

GCF(102, 68) × LCM(102, 68) = 102 × 68
Since the GCF of 102 and 68 = 34
⇒ 34 × LCM(102, 68) = 6936
Therefore, LCM = 204
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How to Find the GCF of 68 and 102 by Long Division Method?

To find the GCF of 68, 102 using long division method, 102 is divided by 68. The corresponding divisor (34) when remainder equals 0 is taken as GCF.

How to Find the GCF of 68 and 102 by Prime Factorization?

To find the GCF of 68 and 102, we will find the prime factorization of the given numbers, i.e. 68 = 2 × 2 × 17; 102 = 2 × 3 × 17.
⇒ Since 2, 17 are common terms in the prime factorization of 68 and 102. Hence, GCF(68, 102) = 2 × 17 = 34
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