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GCF of 27 and 48

GCF of 27 and 48 is the largest possible number that divides 27 and 48 exactly without any remainder. The factors of 27 and 48 are 1, 3, 9, 27 and 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 respectively. There are 3 commonly used methods to find the GCF of 27 and 48 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 27 and 48
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 27 and 48?

Answer: GCF of 27 and 48 is 3.

Explanation:

The GCF of two non-zero integers, x(27) and y(48), is the greatest positive integer m(3) that divides both x(27) and y(48) without any remainder.

Methods to Find GCF of 27 and 48

The methods to find the GCF of 27 and 48 are explained below.

Using Euclid's Algorithm
Long Division Method
Prime Factorization Method

GCF of 27 and 48 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 = 48 and Y = 27

GCF(48, 27) = GCF(27, 48 mod 27) = GCF(27, 21)
GCF(27, 21) = GCF(21, 27 mod 21) = GCF(21, 6)
GCF(21, 6) = GCF(6, 21 mod 6) = GCF(6, 3)
GCF(6, 3) = GCF(3, 6 mod 3) = GCF(3, 0)
GCF(3, 0) = 3 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 27 and 48 is 3.

GCF of 27 and 48 by Long Division

GCF of 27 and 48 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 48 (larger number) by 27 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (27) by the remainder (21).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the GCF of 27 and 48.

GCF of 27 and 48 by Prime Factorization

Prime factorization of 27 and 48 is (3 × 3 × 3) and (2 × 2 × 2 × 2 × 3) respectively. As visible, 27 and 48 have only one common prime factor i.e. 3. Hence, the GCF of 27 and 48 is 3.

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GCF of 27 and 48 Examples

Example 1: Find the greatest number that divides 27 and 48 exactly.

Solution:

The greatest number that divides 27 and 48 exactly is their greatest common factor, i.e. GCF of 27 and 48.
⇒ Factors of 27 and 48:
- Factors of 27 = 1, 3, 9, 27
- Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Therefore, the GCF of 27 and 48 is 3.
Example 2: Find the GCF of 27 and 48, if their LCM is 432.

Solution:

∵ LCM × GCF = 27 × 48
⇒ GCF(27, 48) = (27 × 48)/432 = 3
Therefore, the greatest common factor of 27 and 48 is 3.
Example 3: For two numbers, GCF = 3 and LCM = 432. If one number is 48, find the other number.

Solution:

Given: GCF (x, 48) = 3 and LCM (x, 48) = 432
∵ GCF × LCM = 48 × (x)
⇒ x = (GCF × LCM)/48
⇒ x = (3 × 432)/48
⇒ x = 27
Therefore, the other number is 27.

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FAQs on GCF of 27 and 48

What is the GCF of 27 and 48?

The GCF of 27 and 48 is 3. To calculate the GCF of 27 and 48, we need to factor each number (factors of 27 = 1, 3, 9, 27; factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48) and choose the greatest factor that exactly divides both 27 and 48, i.e., 3.

What is the Relation Between LCM and GCF of 27, 48?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 27 and 48, i.e. GCF × LCM = 27 × 48.

What are the Methods to Find GCF of 27 and 48?

There are three commonly used methods to find the GCF of 27 and 48.

By Long Division
By Prime Factorization
By Euclidean Algorithm

How to Find the GCF of 27 and 48 by Long Division Method?

To find the GCF of 27, 48 using long division method, 48 is divided by 27. The corresponding divisor (3) when remainder equals 0 is taken as GCF.

If the GCF of 48 and 27 is 3, Find its LCM.

GCF(48, 27) × LCM(48, 27) = 48 × 27
Since the GCF of 48 and 27 = 3
⇒ 3 × LCM(48, 27) = 1296
Therefore, LCM = 432
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How to Find the GCF of 27 and 48 by Prime Factorization?

To find the GCF of 27 and 48, we will find the prime factorization of the given numbers, i.e. 27 = 3 × 3 × 3; 48 = 2 × 2 × 2 × 2 × 3.
⇒ Since 3 is the only common prime factor of 27 and 48. Hence, GCF (27, 48) = 3.
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