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GCF of 24 and 84

GCF of 24 and 84 is the largest possible number that divides 24 and 84 exactly without any remainder. The factors of 24 and 84 are 1, 2, 3, 4, 6, 8, 12, 24 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 24 and 84 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 24 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 24 and 84?

Answer: GCF of 24 and 84 is 12.

Explanation:

The GCF of two non-zero integers, x(24) and y(84), is the greatest positive integer m(12) that divides both x(24) and y(84) without any remainder.

Methods to Find GCF of 24 and 84

The methods to find the GCF of 24 and 84 are explained below.

Listing Common Factors
Long Division Method
Using Euclid's Algorithm

GCF of 24 and 84 by Listing Common Factors

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

There are 6 common factors of 24 and 84, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 24 and 84 is 12.

GCF of 24 and 84 by Long Division

GCF of 24 and 84 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 84 (larger number) by 24 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (24) by the remainder (12).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the GCF of 24 and 84.

GCF of 24 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 = 24

GCF(84, 24) = GCF(24, 84 mod 24) = GCF(24, 12)
GCF(24, 12) = GCF(12, 24 mod 12) = GCF(12, 0)
GCF(12, 0) = 12 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 24 and 84 is 12.

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GCF of 24 and 84 Examples

Example 1: For two numbers, GCF = 12 and LCM = 168. If one number is 24, find the other number.

Solution:

Given: GCF (y, 24) = 12 and LCM (y, 24) = 168
∵ GCF × LCM = 24 × (y)
⇒ y = (GCF × LCM)/24
⇒ y = (12 × 168)/24
⇒ y = 84
Therefore, the other number is 84.
Example 2: Find the greatest number that divides 24 and 84 exactly.

Solution:

The greatest number that divides 24 and 84 exactly is their greatest common factor, i.e. GCF of 24 and 84.
⇒ Factors of 24 and 84:
- Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Therefore, the GCF of 24 and 84 is 12.
Example 3: Find the GCF of 24 and 84, if their LCM is 168.

Solution:

∵ LCM × GCF = 24 × 84
⇒ GCF(24, 84) = (24 × 84)/168 = 12
Therefore, the greatest common factor of 24 and 84 is 12.

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FAQs on GCF of 24 and 84

What is the GCF of 24 and 84?

The GCF of 24 and 84 is 12. To calculate the GCF of 24 and 84, we need to factor each number (factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24; factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and choose the greatest factor that exactly divides both 24 and 84, i.e., 12.

What is the Relation Between LCM and GCF of 24, 84?

The following equation can be used to express the relation between LCM and GCF of 24 and 84, i.e. GCF × LCM = 24 × 84.

If the GCF of 84 and 24 is 12, Find its LCM.

GCF(84, 24) × LCM(84, 24) = 84 × 24
Since the GCF of 84 and 24 = 12
⇒ 12 × LCM(84, 24) = 2016
Therefore, LCM = 168
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How to Find the GCF of 24 and 84 by Long Division Method?

To find the GCF of 24, 84 using long division method, 84 is divided by 24. The corresponding divisor (12) when remainder equals 0 is taken as GCF.

What are the Methods to Find GCF of 24 and 84?

There are three commonly used methods to find the GCF of 24 and 84.

By Prime Factorization
By Listing Common Factors
By Long Division

How to Find the GCF of 24 and 84 by Prime Factorization?

To find the GCF of 24 and 84, we will find the prime factorization of the given numbers, i.e. 24 = 2 × 2 × 2 × 3; 84 = 2 × 2 × 3 × 7.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 24 and 84. Hence, GCF(24, 84) = 2 × 2 × 3 = 12
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