Greatest Common Divisor (GCD): Learn Definition, Steps & Methods to Find GCD using Examples!

What is the Greatest Common Divisor?

The largest possible positive integer that divides the provided numbers and leaves zero as the remainder is known as the greatest common divisor in mathematics. This is how we define the GCD. The greatest common denominator, greatest common factor (GCF), or highest common factor are all terms used to describe the Highest Common Factor. When we talk about the Greatest Common Divisor, we can easily replace the adjective ‘greatest’ with ‘highest, and the word ‘divisor’ with ‘factor’. As a result, the GCD is also known as the Highest Common Factor, Greatest Common Factor (GCF), and so on. For example, the GCD of 8 and 12 is 4.

Steps to Find Greatest Common Divisor

As we have established, the GCD of any two or more such integers will be the largest integer that will divide each of the integers such that their remains will be zero. So, there are various methods or algorithms to determine the GCD (Greatest Common Divisor) between any two given numbers. So, if we talk about the easiest and fastest process to calculate the GCD, it would consist of the following steps:

• Step 1 : Decompose every one of the numbers given in the form of products of prime factors.
• Step 2 : Successively dividing each one of the numbers by the prime numbers until we reach a quotient that equals 1. This is called prime factorization.
• Step 3 : Once, we have all the prime factors of the numbers we have to find the highest common factor or divisors.

Methods to Find GCD

There are two methods to find the Greatest Common Divisor GCD. They are as listed below:

• LCM Method:

The LCM Method helps find the Greatest Common Divisor (GCD) of two numbers, a and b. It involves multiplying a and b, then dividing the result by their least common multiple (LCM). In short, GCD(a, b) = (a × b) / LCM(a, b). This method streamlines the GCD calculation through multiplication and division.

• Euclid’s Algorithm:

Euclid's Algorithm, also known as the Euclidean Algorithm, is an efficient method to find the Greatest Common Divisor (GCD) of two positive integers. It works by replacing the bigger number with the remainder obtained by dividing it by the smaller number repeatedly until the remainder becomes zero. This process helps find the Greatest Common Divisor (GCD).

LCM Method

The GCD of two positive integers (a, b) can be determined using the LCM Method with the GCD formula:

To calculate the GCD of (a, b) using the LCM method, follow these steps:

• Step 1: Calculate the product of a and b.
• Step 2: Find the least common multiple of a and b.
• Step 3: Divide the results from Steps 1 and 2.
• Step 4: The acquired value by division is the greatest common divisor of (a, b).

Euclid’s Algorithm

We can use the use of Euclid’s Division Lemma or the Euclid Division Algorithm to get the GCD. It is a commonly used method by all students. Here, we’ll look at how to use Euclid’s Division Lemma Algorithm to find the GCD of two or more numbers. It is also called as greatest common divisor algorithm. Follow the below steps to find the GCD of given numbers with Euclid’s Division Lemma:

Step 1: Apply Euclid’s division lemma, to a and b. So, we find whole numbers, q and r such that a = bq + r, 0 ≤ r < b.

Step 2: If r = 0, b is the GCD of a and b. If r ≠ 0, apply the division lemma to b and r.

Step 3: Continue the process until the remainder is zero. The divisor at this stage will be the required GCD of a and b.

Thus, Euclid’s Division Lemma algorithm works because GCD (a, b) = GCD (b, r) where the symbol GCD (a, b) denotes the GCD of a and b.

Solved Examples of Greatest Common Divisor

Let’s see some solved examples on Greatest Common Divisor.

Solved Example 1: Find the greatest common divisor (GCD) of 70, 210 and 315?

Solution:

Factors of 70 = 1, 2, 5, 7, 10, 14, 35, and 70.

Factors of 210 =1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210.

Factors of 315 =1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, and 315. Therefore, common factor of 70, 210 and 315 = 1,5, 7 and 35.

Greatest common divisor (GCD) of 70, 210 and 315 = 35.

Solved Example 2: Use the Euclidean Algorithm to find the greatest common divisor of 44 and 17.

Solution:

The Euclidean Algorithm yields:

44 = 2 x 17 + 10

17 = 1 x 10 + 7

10 = 1 x 7 + 3

7 = 2 x 3 + 1.

Therefore the greatest common divisor of 44 and 17 is 1

Solved Example 3: Finding GCD of 560, 570, 265 using Factoring.

Solution:

To find the GCD of numbers using factoring list out all the divisors of each number

Divisors of 560

List of positive integer divisors of 560 that divides 560 without a remainder.

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560

Divisors of 570

List of positive integer divisors of 570 that divides 570 without a remainder.

1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570

Divisors of 265

List of positive integer divisors of 265 that divides 265 without a remainder.

1, 5, 53, 265

GCD

We found the divisors of 560, 570, 265. The biggest common divisor number is the GCD number. So the Greatest Common Divisor 560, 570, 265 is 5.

Solved Example 4: Finding GCD of 560, 570, 265 using LCM Formula.

Solution:

Step1: Let’s calculate the GCD of first two numbers. The formula of GCD is GCD(a, b) = ( a x b) / LCM(a, b)

LCM(560, 570) = 31920

GCD(560, 570) = ( 560 x 570 ) / 31920

GCD(560, 570) = 319200 / 31920

GCD(560, 570) = 10

Step 2: Here we consider the GCD from the above i.e. 10 as the first number and the next as 265. The formula of GCD is GCD(a, b) = ( a x b) / LCM(a, b)

LCM(10, 265) = 530

GCD(10, 265) = ( 10 x 265 ) / 530

GCD(10, 265) = 2650 / 530

GCD(10, 265) = 5

GCD of 560, 570, 265 is 5

Questions to Ace Your Exams

Q1. How to find the greatest common divisor? How to find the greatest common divisor of two numbers? How to calculate the greatest common divisor?
The process of determining the greatest common divisor (GCD) involves employing the Euclidean algorithm, a fundamental method in number theory. Begin by finding the remainder when dividing the larger number by the smaller one. Subsequently, replace the larger number with the smaller and the smaller number with the remainder. Iteratively continue this process until the remainder becomes zero, and the last non-zero remainder is identified as the GCD.

Q2. What is the greatest common divisor of 24 and 32? How to find the greatest common divisor of two numbers? What is the greatest common divisor of 378 and 420? What is the greatest common divisor of 63 and 81?
The greatest common divisor (GCD) of 24 and 32 is determined by applying the prime factorization method. Expressing both numbers as products of prime factors, the common primes are identified, resulting in a GCD of 8.
To find the GCD of two numbers, use prime factorization or the Euclidean algorithm. Prime factorization involves expressing each number as a product of prime factors and identifying the common primes, ultimately yielding the GCD.
For 378 and 420, employing prime factorization reveals common primes, leading to a GCD of 42.
Similarly, the GCD of 63 and 81, computed through prime factorization, is found to be 9.

Q3. How to find the greatest common divisor in C ? How to find the greatest common divisor in Java?
In C, you can find the greatest common divisor (GCD) using the Euclidean algorithm, which involves repeatedly applying the formula: GCD(a, b) = GCD(b, a % b) until the remainder becomes zero. This iterative process efficiently determines the GCD.
In Java, you can use a very similar approach to the Euclidean algorithm for GCD calculation. Iterate through the algorithm until the remainder becomes zero, providing an effective way to find the GCD of two numbers.
Both C and Java commonly use loops or recursive functions to implement the Euclidean algorithm for GCD computation.

Q4. How to find the greatest common divisor in Python?
In Python, determining the greatest common divisor (GCD) of two numbers is simplified with the math module, which includes a dedicated gcd(a, b) function. This function employs the efficient Euclidean algorithm, providing a concise solution without the necessity to implement the algorithm manually. By using math.gcd(a, b), Python programmers can effortlessly calculate the GCD of two numbers. This built-in functionality not only streamlines the code but also ensures a reliable and optimized approach to GCD computations, aligning with Python's commitment to readability and convenience in programming tasks.

Conclusion

In a nutshell, mastering the Greatest Common Divisor (GCD) is a game-changer for simplifying math problems, especially on tests like the SAT, ACT, GED, and AP exams. Whether you’re using prime factorization, the LCM method, or Euclid’s Algorithm, knowing how to find the GCD quickly gives you a serious edge. Practice a few examples, and you’ll be ready to tackle any GCD question that comes your way!