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GCF of 56 and 21

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

1.	GCF of 56 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 56 and 21?

Answer: GCF of 56 and 21 is 7.

Explanation:

The GCF of two non-zero integers, x(56) and y(21), is the greatest positive integer m(7) that divides both x(56) and y(21) without any remainder.

Methods to Find GCF of 56 and 21

Let's look at the different methods for finding the GCF of 56 and 21.

Prime Factorization Method
Long Division Method
Using Euclid's Algorithm

GCF of 56 and 21 by Prime Factorization

Prime factorization of 56 and 21 is (2 × 2 × 2 × 7) and (3 × 7) respectively. As visible, 56 and 21 have only one common prime factor i.e. 7. Hence, the GCF of 56 and 21 is 7.

GCF of 56 and 21 by Long Division

GCF of 56 and 21 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 56 (larger number) by 21 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (21) by the remainder (14).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (7) is the GCF of 56 and 21.

GCF of 56 and 21 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 = 56 and Y = 21

GCF(56, 21) = GCF(21, 56 mod 21) = GCF(21, 14)
GCF(21, 14) = GCF(14, 21 mod 14) = GCF(14, 7)
GCF(14, 7) = GCF(7, 14 mod 7) = GCF(7, 0)
GCF(7, 0) = 7 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 56 and 21 is 7.

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GCF of 56 and 21 Examples

Example 1: For two numbers, GCF = 7 and LCM = 168. If one number is 21, find the other number.

Solution:

Given: GCF (y, 21) = 7 and LCM (y, 21) = 168
∵ GCF × LCM = 21 × (y)
⇒ y = (GCF × LCM)/21
⇒ y = (7 × 168)/21
⇒ y = 56
Therefore, the other number is 56.
Example 2: The product of two numbers is 1176. If their GCF is 7, what is their LCM?

Solution:

Given: GCF = 7 and product of numbers = 1176
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 1176/7
Therefore, the LCM is 168.
Example 3: Find the greatest number that divides 56 and 21 exactly.

Solution:

The greatest number that divides 56 and 21 exactly is their greatest common factor, i.e. GCF of 56 and 21.
⇒ Factors of 56 and 21:
- Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56
- Factors of 21 = 1, 3, 7, 21
Therefore, the GCF of 56 and 21 is 7.

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FAQs on GCF of 56 and 21

What is the GCF of 56 and 21?

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

What are the Methods to Find GCF of 56 and 21?

There are three commonly used methods to find the GCF of 56 and 21.

By Long Division
By Prime Factorization
By Euclidean Algorithm

What is the Relation Between LCM and GCF of 56, 21?

The following equation can be used to express the relation between Least Common Multiple and GCF of 56 and 21, i.e. GCF × LCM = 56 × 21.

How to Find the GCF of 56 and 21 by Prime Factorization?

To find the GCF of 56 and 21, we will find the prime factorization of the given numbers, i.e. 56 = 2 × 2 × 2 × 7; 21 = 3 × 7.
⇒ Since 7 is the only common prime factor of 56 and 21. Hence, GCF (56, 21) = 7.
☛ What are Prime Numbers?

How to Find the GCF of 56 and 21 by Long Division Method?

To find the GCF of 56, 21 using long division method, 56 is divided by 21. The corresponding divisor (7) when remainder equals 0 is taken as GCF.

If the GCF of 21 and 56 is 7, Find its LCM.

GCF(21, 56) × LCM(21, 56) = 21 × 56
Since the GCF of 21 and 56 = 7
⇒ 7 × LCM(21, 56) = 1176
Therefore, LCM = 168
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