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GCF of 65 and 91

GCF of 65 and 91 is the largest possible number that divides 65 and 91 exactly without any remainder. The factors of 65 and 91 are 1, 5, 13, 65 and 1, 7, 13, 91 respectively. There are 3 commonly used methods to find the GCF of 65 and 91 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 65 and 91
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 65 and 91?

Answer: GCF of 65 and 91 is 13.

Explanation:

The GCF of two non-zero integers, x(65) and y(91), is the greatest positive integer m(13) that divides both x(65) and y(91) without any remainder.

Methods to Find GCF of 65 and 91

The methods to find the GCF of 65 and 91 are explained below.

Long Division Method
Prime Factorization Method
Using Euclid's Algorithm

GCF of 65 and 91 by Long Division

GCF of 65 and 91 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 91 (larger number) by 65 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (65) by the remainder (26).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (13) is the GCF of 65 and 91.

GCF of 65 and 91 by Prime Factorization

Prime factorization of 65 and 91 is (5 × 13) and (7 × 13) respectively. As visible, 65 and 91 have only one common prime factor i.e. 13. Hence, the GCF of 65 and 91 is 13.

GCF of 65 and 91 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 = 91 and Y = 65

GCF(91, 65) = GCF(65, 91 mod 65) = GCF(65, 26)
GCF(65, 26) = GCF(26, 65 mod 26) = GCF(26, 13)
GCF(26, 13) = GCF(13, 26 mod 13) = GCF(13, 0)
GCF(13, 0) = 13 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 65 and 91 is 13.

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GCF of 65 and 91 Examples

Example 1: Find the GCF of 65 and 91, if their LCM is 455.

Solution:

∵ LCM × GCF = 65 × 91
⇒ GCF(65, 91) = (65 × 91)/455 = 13
Therefore, the greatest common factor of 65 and 91 is 13.
Example 2: Find the greatest number that divides 65 and 91 exactly.

Solution:

The greatest number that divides 65 and 91 exactly is their greatest common factor, i.e. GCF of 65 and 91.
⇒ Factors of 65 and 91:
- Factors of 65 = 1, 5, 13, 65
- Factors of 91 = 1, 7, 13, 91
Therefore, the GCF of 65 and 91 is 13.
Example 3: The product of two numbers is 5915. If their GCF is 13, what is their LCM?

Solution:

Given: GCF = 13 and product of numbers = 5915
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 5915/13
Therefore, the LCM is 455.

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FAQs on GCF of 65 and 91

What is the GCF of 65 and 91?

The GCF of 65 and 91 is 13. To calculate the greatest common factor (GCF) of 65 and 91, we need to factor each number (factors of 65 = 1, 5, 13, 65; factors of 91 = 1, 7, 13, 91) and choose the greatest factor that exactly divides both 65 and 91, i.e., 13.

What are the Methods to Find GCF of 65 and 91?

There are three commonly used methods to find the GCF of 65 and 91.

By Euclidean Algorithm
By Long Division
By Prime Factorization

How to Find the GCF of 65 and 91 by Long Division Method?

To find the GCF of 65, 91 using long division method, 91 is divided by 65. The corresponding divisor (13) when remainder equals 0 is taken as GCF.

If the GCF of 91 and 65 is 13, Find its LCM.

GCF(91, 65) × LCM(91, 65) = 91 × 65
Since the GCF of 91 and 65 = 13
⇒ 13 × LCM(91, 65) = 5915
Therefore, LCM = 455
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What is the Relation Between LCM and GCF of 65, 91?

The following equation can be used to express the relation between LCM and GCF of 65 and 91, i.e. GCF × LCM = 65 × 91.

How to Find the GCF of 65 and 91 by Prime Factorization?

To find the GCF of 65 and 91, we will find the prime factorization of the given numbers, i.e. 65 = 5 × 13; 91 = 7 × 13.
⇒ Since 13 is the only common prime factor of 65 and 91. Hence, GCF (65, 91) = 13.
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