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GCF of 72 and 84

GCF of 72 and 84 is the largest possible number that divides 72 and 84 exactly without any remainder. The factors of 72 and 84 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 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 72 and 84 - prime factorization, Euclidean algorithm, and long division.

1.	GCF of 72 and 84
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 72 and 84?

Answer: GCF of 72 and 84 is 12.

Explanation:

The GCF of two non-zero integers, x(72) and y(84), is the greatest positive integer m(12) that divides both x(72) and y(84) without any remainder.

Methods to Find GCF of 72 and 84

The methods to find the GCF of 72 and 84 are explained below.

Prime Factorization Method
Using Euclid's Algorithm
Long Division Method

GCF of 72 and 84 by Prime Factorization

Prime factorization of 72 and 84 is (2 × 2 × 2 × 3 × 3) and (2 × 2 × 3 × 7) respectively. As visible, 72 and 84 have common prime factors. Hence, the GCF of 72 and 84 is 2 × 2 × 3 = 12.

GCF of 72 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 = 72

GCF(84, 72) = GCF(72, 84 mod 72) = GCF(72, 12)
GCF(72, 12) = GCF(12, 72 mod 12) = GCF(12, 0)
GCF(12, 0) = 12 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 72 and 84 is 12.

GCF of 72 and 84 by Long Division

GCF of 72 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 72 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (72) by the remainder (12).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (12) is the GCF of 72 and 84.

☛ Also Check:

GCF of 72 and 84 Examples

Example 1: Find the GCF of 72 and 84, if their LCM is 504.

Solution:

∵ LCM × GCF = 72 × 84
⇒ GCF(72, 84) = (72 × 84)/504 = 12
Therefore, the greatest common factor of 72 and 84 is 12.
Example 2: Find the greatest number that divides 72 and 84 exactly.

Solution:

The greatest number that divides 72 and 84 exactly is their greatest common factor, i.e. GCF of 72 and 84.
⇒ Factors of 72 and 84:
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Therefore, the GCF of 72 and 84 is 12.
Example 3: For two numbers, GCF = 12 and LCM = 504. If one number is 72, find the other number.

Solution:

Given: GCF (z, 72) = 12 and LCM (z, 72) = 504
∵ GCF × LCM = 72 × (z)
⇒ z = (GCF × LCM)/72
⇒ z = (12 × 504)/72
⇒ z = 84
Therefore, the other number is 84.

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FAQs on GCF of 72 and 84

What is the GCF of 72 and 84?

The GCF of 72 and 84 is 12. To calculate the greatest common factor (GCF) of 72 and 84, we need to factor each number (factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84) and choose the greatest factor that exactly divides both 72 and 84, i.e., 12.

How to Find the GCF of 72 and 84 by Long Division Method?

To find the GCF of 72, 84 using long division method, 84 is divided by 72. The corresponding divisor (12) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 72, 84?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 72 and 84, i.e. GCF × LCM = 72 × 84.

How to Find the GCF of 72 and 84 by Prime Factorization?

To find the GCF of 72 and 84, we will find the prime factorization of the given numbers, i.e. 72 = 2 × 2 × 2 × 3 × 3; 84 = 2 × 2 × 3 × 7.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 72 and 84. Hence, GCF(72, 84) = 2 × 2 × 3 = 12
☛ What is a Prime Number?

What are the Methods to Find GCF of 72 and 84?

There are three commonly used methods to find the GCF of 72 and 84.

By Listing Common Factors
By Prime Factorization
By Long Division

If the GCF of 84 and 72 is 12, Find its LCM.

GCF(84, 72) × LCM(84, 72) = 84 × 72
Since the GCF of 84 and 72 = 12
⇒ 12 × LCM(84, 72) = 6048
Therefore, LCM = 504
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