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GCF of 37 and 67

GCF of 37 and 67 is the largest possible number that divides 37 and 67 exactly without any remainder. The factors of 37 and 67 are 1, 37 and 1, 67 respectively. There are 3 commonly used methods to find the GCF of 37 and 67 - long division, Euclidean algorithm, and prime factorization.

1.	GCF of 37 and 67
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 37 and 67?

Answer: GCF of 37 and 67 is 1.

Explanation:

The GCF of two non-zero integers, x(37) and y(67), is the greatest positive integer m(1) that divides both x(37) and y(67) without any remainder.

Methods to Find GCF of 37 and 67

Let's look at the different methods for finding the GCF of 37 and 67.

Prime Factorization Method
Using Euclid's Algorithm
Long Division Method

GCF of 37 and 67 by Prime Factorization

Prime factorization of 37 and 67 is (37) and (67) respectively. As visible, there are no common prime factors between 37 and 67, i.e. they are coprime. Hence, the GCF of 37 and 67 will be 1.

GCF of 37 and 67 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 = 67 and Y = 37

GCF(67, 37) = GCF(37, 67 mod 37) = GCF(37, 30)
GCF(37, 30) = GCF(30, 37 mod 30) = GCF(30, 7)
GCF(30, 7) = GCF(7, 30 mod 7) = GCF(7, 2)
GCF(7, 2) = GCF(2, 7 mod 2) = GCF(2, 1)
GCF(2, 1) = 1 (∵ GCF(X, 1) = 1)

Therefore, the value of GCF of 37 and 67 is 1.

GCF of 37 and 67 by Long Division

GCF of 37 and 67 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 67 (larger number) by 37 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (37) by the remainder (30).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (1) is the GCF of 37 and 67.

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GCF of 37 and 67 Examples

Example 1: Find the greatest number that divides 37 and 67 exactly.

Solution:

The greatest number that divides 37 and 67 exactly is their greatest common factor, i.e. GCF of 37 and 67.
⇒ Factors of 37 and 67:
- Factors of 37 = 1, 37
- Factors of 67 = 1, 67
Therefore, the GCF of 37 and 67 is 1.
Example 2: For two numbers, GCF = 1 and LCM = 2479. If one number is 67, find the other number.

Solution:

Given: GCF (z, 67) = 1 and LCM (z, 67) = 2479
∵ GCF × LCM = 67 × (z)
⇒ z = (GCF × LCM)/67
⇒ z = (1 × 2479)/67
⇒ z = 37
Therefore, the other number is 37.
Example 3: The product of two numbers is 2479. If their GCF is 1, what is their LCM?

Solution:

Given: GCF = 1 and product of numbers = 2479
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 2479/1
Therefore, the LCM is 2479.

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FAQs on GCF of 37 and 67

What is the GCF of 37 and 67?

The GCF of 37 and 67 is 1. To calculate the greatest common factor (GCF) of 37 and 67, we need to factor each number (factors of 37 = 1, 37; factors of 67 = 1, 67) and choose the greatest factor that exactly divides both 37 and 67, i.e., 1.

What is the Relation Between LCM and GCF of 37, 67?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 37 and 67, i.e. GCF × LCM = 37 × 67.

What are the Methods to Find GCF of 37 and 67?

There are three commonly used methods to find the GCF of 37 and 67.

By Long Division
By Listing Common Factors
By Prime Factorization

How to Find the GCF of 37 and 67 by Prime Factorization?

To find the GCF of 37 and 67, we will find the prime factorization of the given numbers, i.e. 37 = 37; 67 = 67.
⇒ There is no common prime factor for 37 and 67. Hence, GCF (37, 67) = 1.
☛ Prime Numbers

How to Find the GCF of 37 and 67 by Long Division Method?

To find the GCF of 37, 67 using long division method, 67 is divided by 37. The corresponding divisor (1) when remainder equals 0 is taken as GCF.

If the GCF of 67 and 37 is 1, Find its LCM.

GCF(67, 37) × LCM(67, 37) = 67 × 37
Since the GCF of 67 and 37 = 1
⇒ 1 × LCM(67, 37) = 2479
Therefore, LCM = 2479
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