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GCF of 84 and 90

GCF of 84 and 90 is the largest possible number that divides 84 and 90 exactly without any remainder. The factors of 84 and 90 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 respectively. There are 3 commonly used methods to find the GCF of 84 and 90 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 84 and 90
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 84 and 90?

Answer: GCF of 84 and 90 is 6.

Explanation:

The GCF of two non-zero integers, x(84) and y(90), is the greatest positive integer m(6) that divides both x(84) and y(90) without any remainder.

Methods to Find GCF of 84 and 90

The methods to find the GCF of 84 and 90 are explained below.

Using Euclid's Algorithm
Long Division Method
Prime Factorization Method

GCF of 84 and 90 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 = 90 and Y = 84

GCF(90, 84) = GCF(84, 90 mod 84) = GCF(84, 6)
GCF(84, 6) = GCF(6, 84 mod 6) = GCF(6, 0)
GCF(6, 0) = 6 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 84 and 90 is 6.

GCF of 84 and 90 by Long Division

GCF of 84 and 90 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 90 (larger number) by 84 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (6).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (6) is the GCF of 84 and 90.

GCF of 84 and 90 by Prime Factorization

Prime factorization of 84 and 90 is (2 × 2 × 3 × 7) and (2 × 3 × 3 × 5) respectively. As visible, 84 and 90 have common prime factors. Hence, the GCF of 84 and 90 is 2 × 3 = 6.

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GCF of 84 and 90 Examples

Example 1: Find the GCF of 84 and 90, if their LCM is 1260.

Solution:

∵ LCM × GCF = 84 × 90
⇒ GCF(84, 90) = (84 × 90)/1260 = 6
Therefore, the greatest common factor of 84 and 90 is 6.
Example 2: Find the greatest number that divides 84 and 90 exactly.

Solution:

The greatest number that divides 84 and 90 exactly is their greatest common factor, i.e. GCF of 84 and 90.
⇒ Factors of 84 and 90:
- Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Therefore, the GCF of 84 and 90 is 6.
Example 3: For two numbers, GCF = 6 and LCM = 1260. If one number is 90, find the other number.

Solution:

Given: GCF (z, 90) = 6 and LCM (z, 90) = 1260
∵ GCF × LCM = 90 × (z)
⇒ z = (GCF × LCM)/90
⇒ z = (6 × 1260)/90
⇒ z = 84
Therefore, the other number is 84.

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FAQs on GCF of 84 and 90

What is the GCF of 84 and 90?

The GCF of 84 and 90 is 6. To calculate the greatest common factor (GCF) of 84 and 90, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90) and choose the greatest factor that exactly divides both 84 and 90, i.e., 6.

What is the Relation Between LCM and GCF of 84, 90?

The following equation can be used to express the relation between Least Common Multiple and GCF of 84 and 90, i.e. GCF × LCM = 84 × 90.

If the GCF of 90 and 84 is 6, Find its LCM.

GCF(90, 84) × LCM(90, 84) = 90 × 84
Since the GCF of 90 and 84 = 6
⇒ 6 × LCM(90, 84) = 7560
Therefore, LCM = 1260
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How to Find the GCF of 84 and 90 by Prime Factorization?

To find the GCF of 84 and 90, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 90 = 2 × 3 × 3 × 5.
⇒ Since 2, 3 are common terms in the prime factorization of 84 and 90. Hence, GCF(84, 90) = 2 × 3 = 6
☛ What is a Prime Number?

What are the Methods to Find GCF of 84 and 90?

There are three commonly used methods to find the GCF of 84 and 90.

By Listing Common Factors
By Long Division
By Prime Factorization

How to Find the GCF of 84 and 90 by Long Division Method?

To find the GCF of 84, 90 using long division method, 90 is divided by 84. The corresponding divisor (6) when remainder equals 0 is taken as GCF.

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