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GCF of 56 and 84

GCF of 56 and 84 is the largest possible number that divides 56 and 84 exactly without any remainder. The factors of 56 and 84 are 1, 2, 4, 7, 8, 14, 28, 56 and 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 respectively. There are 3 commonly used methods to find the GCF of 56 and 84 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 56 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 56 and 84?

Answer: GCF of 56 and 84 is 28.

Explanation:

The GCF of two non-zero integers, x(56) and y(84), is the greatest positive integer m(28) that divides both x(56) and y(84) without any remainder.

Methods to Find GCF of 56 and 84

The methods to find the GCF of 56 and 84 are explained below.

Prime Factorization Method
Listing Common Factors
Using Euclid's Algorithm

GCF of 56 and 84 by Prime Factorization

Prime factorization of 56 and 84 is (2 × 2 × 2 × 7) and (2 × 2 × 3 × 7) respectively. As visible, 56 and 84 have common prime factors. Hence, the GCF of 56 and 84 is 2 × 2 × 7 = 28.

GCF of 56 and 84 by Listing Common Factors

Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

There are 6 common factors of 56 and 84, that are 1, 2, 4, 7, 14, and 28. Therefore, the greatest common factor of 56 and 84 is 28.

GCF of 56 and 84 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 = 84 and Y = 56

GCF(84, 56) = GCF(56, 84 mod 56) = GCF(56, 28)
GCF(56, 28) = GCF(28, 56 mod 28) = GCF(28, 0)
GCF(28, 0) = 28 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 56 and 84 is 28.

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GCF of 56 and 84 Examples

Example 1: The product of two numbers is 4704. If their GCF is 28, what is their LCM?

Solution:

Given: GCF = 28 and product of numbers = 4704
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 4704/28
Therefore, the LCM is 168.
Example 2: For two numbers, GCF = 28 and LCM = 168. If one number is 56, find the other number.

Solution:

Given: GCF (x, 56) = 28 and LCM (x, 56) = 168
∵ GCF × LCM = 56 × (x)
⇒ x = (GCF × LCM)/56
⇒ x = (28 × 168)/56
⇒ x = 84
Therefore, the other number is 84.
Example 3: Find the GCF of 56 and 84, if their LCM is 168.

Solution:

∵ LCM × GCF = 56 × 84
⇒ GCF(56, 84) = (56 × 84)/168 = 28
Therefore, the greatest common factor of 56 and 84 is 28.

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FAQs on GCF of 56 and 84

What is the GCF of 56 and 84?

The GCF of 56 and 84 is 28. To calculate the GCF of 56 and 84, we need to factor each number (factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56; factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and choose the greatest factor that exactly divides both 56 and 84, i.e., 28.

How to Find the GCF of 56 and 84 by Prime Factorization?

To find the GCF of 56 and 84, we will find the prime factorization of the given numbers, i.e. 56 = 2 × 2 × 2 × 7; 84 = 2 × 2 × 3 × 7.
⇒ Since 2, 2, 7 are common terms in the prime factorization of 56 and 84. Hence, GCF(56, 84) = 2 × 2 × 7 = 28
☛ Prime Numbers

What are the Methods to Find GCF of 56 and 84?

There are three commonly used methods to find the GCF of 56 and 84.

By Listing Common Factors
By Long Division
By Prime Factorization

If the GCF of 84 and 56 is 28, Find its LCM.

GCF(84, 56) × LCM(84, 56) = 84 × 56
Since the GCF of 84 and 56 = 28
⇒ 28 × LCM(84, 56) = 4704
Therefore, LCM = 168
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How to Find the GCF of 56 and 84 by Long Division Method?

To find the GCF of 56, 84 using long division method, 84 is divided by 56. The corresponding divisor (28) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 56, 84?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 56 and 84, i.e. GCF × LCM = 56 × 84.

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