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GCF of 52 and 65

GCF of 52 and 65 is the largest possible number that divides 52 and 65 exactly without any remainder. The factors of 52 and 65 are 1, 2, 4, 13, 26, 52 and 1, 5, 13, 65 respectively. There are 3 commonly used methods to find the GCF of 52 and 65 - prime factorization, Euclidean algorithm, and long division.

1.	GCF of 52 and 65
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 52 and 65?

Answer: GCF of 52 and 65 is 13.

Explanation:

The GCF of two non-zero integers, x(52) and y(65), is the greatest positive integer m(13) that divides both x(52) and y(65) without any remainder.

Methods to Find GCF of 52 and 65

The methods to find the GCF of 52 and 65 are explained below.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

GCF of 52 and 65 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 = 65 and Y = 52

GCF(65, 52) = GCF(52, 65 mod 52) = GCF(52, 13)
GCF(52, 13) = GCF(13, 52 mod 13) = GCF(13, 0)
GCF(13, 0) = 13 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 52 and 65 is 13.

GCF of 52 and 65 by Prime Factorization

Prime factorization of 52 and 65 is (2 × 2 × 13) and (5 × 13) respectively. As visible, 52 and 65 have only one common prime factor i.e. 13. Hence, the GCF of 52 and 65 is 13.

GCF of 52 and 65 by Long Division

GCF of 52 and 65 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 65 (larger number) by 52 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (52) by the remainder (13).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (13) is the GCF of 52 and 65.

☛ Also Check:

GCF of 52 and 65 Examples

Example 1: For two numbers, GCF = 13 and LCM = 260. If one number is 52, find the other number.

Solution:

Given: GCF (x, 52) = 13 and LCM (x, 52) = 260
∵ GCF × LCM = 52 × (x)
⇒ x = (GCF × LCM)/52
⇒ x = (13 × 260)/52
⇒ x = 65
Therefore, the other number is 65.
Example 2: Find the GCF of 52 and 65, if their LCM is 260.

Solution:

∵ LCM × GCF = 52 × 65
⇒ GCF(52, 65) = (52 × 65)/260 = 13
Therefore, the greatest common factor of 52 and 65 is 13.
Example 3: Find the greatest number that divides 52 and 65 exactly.

Solution:

The greatest number that divides 52 and 65 exactly is their greatest common factor, i.e. GCF of 52 and 65.
⇒ Factors of 52 and 65:
- Factors of 52 = 1, 2, 4, 13, 26, 52
- Factors of 65 = 1, 5, 13, 65
Therefore, the GCF of 52 and 65 is 13.

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FAQs on GCF of 52 and 65

What is the GCF of 52 and 65?

The GCF of 52 and 65 is 13. To calculate the greatest common factor of 52 and 65, we need to factor each number (factors of 52 = 1, 2, 4, 13, 26, 52; factors of 65 = 1, 5, 13, 65) and choose the greatest factor that exactly divides both 52 and 65, i.e., 13.

What are the Methods to Find GCF of 52 and 65?

There are three commonly used methods to find the GCF of 52 and 65.

By Prime Factorization
By Long Division
By Euclidean Algorithm

How to Find the GCF of 52 and 65 by Long Division Method?

To find the GCF of 52, 65 using long division method, 65 is divided by 52. The corresponding divisor (13) when remainder equals 0 is taken as GCF.

How to Find the GCF of 52 and 65 by Prime Factorization?

To find the GCF of 52 and 65, we will find the prime factorization of the given numbers, i.e. 52 = 2 × 2 × 13; 65 = 5 × 13.
⇒ Since 13 is the only common prime factor of 52 and 65. Hence, GCF (52, 65) = 13.
☛ What is a Prime Number?

What is the Relation Between LCM and GCF of 52, 65?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 52 and 65, i.e. GCF × LCM = 52 × 65.

If the GCF of 65 and 52 is 13, Find its LCM.

GCF(65, 52) × LCM(65, 52) = 65 × 52
Since the GCF of 65 and 52 = 13
⇒ 13 × LCM(65, 52) = 3380
Therefore, LCM = 260
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