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GCF of 51 and 68

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

1.	GCF of 51 and 68
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 51 and 68?

Answer: GCF of 51 and 68 is 17.

Explanation:

The GCF of two non-zero integers, x(51) and y(68), is the greatest positive integer m(17) that divides both x(51) and y(68) without any remainder.

Methods to Find GCF of 51 and 68

The methods to find the GCF of 51 and 68 are explained below.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

GCF of 51 and 68 by Prime Factorization

Prime factorization of 51 and 68 is (3 × 17) and (2 × 2 × 17) respectively. As visible, 51 and 68 have only one common prime factor i.e. 17. Hence, the GCF of 51 and 68 is 17.

GCF of 51 and 68 by Long Division

GCF of 51 and 68 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 68 (larger number) by 51 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (51) by the remainder (17).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (17) is the GCF of 51 and 68.

GCF of 51 and 68 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 = 68 and Y = 51

GCF(68, 51) = GCF(51, 68 mod 51) = GCF(51, 17)
GCF(51, 17) = GCF(17, 51 mod 17) = GCF(17, 0)
GCF(17, 0) = 17 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 51 and 68 is 17.

☛ Also Check:

GCF of 51 and 68 Examples

Example 1: Find the GCF of 51 and 68, if their LCM is 204.

Solution:

∵ LCM × GCF = 51 × 68
⇒ GCF(51, 68) = (51 × 68)/204 = 17
Therefore, the greatest common factor of 51 and 68 is 17.
Example 2: The product of two numbers is 3468. If their GCF is 17, what is their LCM?

Solution:

Given: GCF = 17 and product of numbers = 3468
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 3468/17
Therefore, the LCM is 204.
Example 3: Find the greatest number that divides 51 and 68 exactly.

Solution:

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

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FAQs on GCF of 51 and 68

What is the GCF of 51 and 68?

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

How to Find the GCF of 51 and 68 by Long Division Method?

To find the GCF of 51, 68 using long division method, 68 is divided by 51. The corresponding divisor (17) when remainder equals 0 is taken as GCF.

How to Find the GCF of 51 and 68 by Prime Factorization?

To find the GCF of 51 and 68, we will find the prime factorization of the given numbers, i.e. 51 = 3 × 17; 68 = 2 × 2 × 17.
⇒ Since 17 is the only common prime factor of 51 and 68. Hence, GCF (51, 68) = 17.
☛ Prime Numbers

What is the Relation Between LCM and GCF of 51, 68?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 51 and 68, i.e. GCF × LCM = 51 × 68.

What are the Methods to Find GCF of 51 and 68?

There are three commonly used methods to find the GCF of 51 and 68.

By Prime Factorization
By Long Division
By Listing Common Factors

If the GCF of 68 and 51 is 17, Find its LCM.

GCF(68, 51) × LCM(68, 51) = 68 × 51
Since the GCF of 68 and 51 = 17
⇒ 17 × LCM(68, 51) = 3468
Therefore, LCM = 204
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