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Explore the concept of the Greatest Common Divisor (GCD) in this comprehensive guide. Learn the definition, step-by-step methods, and practical examples to find GCD efficiently. Delve into prime factorization, LCM Method, and Euclid’s Algorithm for a clear understanding.
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The Greatest Common Divisor (GCD) of two or more integers, when at least one of them is not zero, is the largest positive integer that is a divisor of both numbers. The greatest common divisor is also known as the greatest common factor (GCF), highest common factor (HCF), greatest common measure (GCM), or highest common divisor. A common divisor or factor is a number that divides two or more non-zero numbers. It is different for different pairs or groups of numbers. It is always greater than 0. In this article, we will learn about GCD, steps and methods to find GCD with solved examples and FAQs.
The Greatest Common Divisor (GCD) is the largest positive number that can divide two or more numbers exactly, without leaving any remainder. In other words, it is the biggest number that all the given numbers share as a factor. The GCD is also called the Greatest Common Factor (GCF) or the Highest Common Factor (HCF) — all of these terms mean the same thing. The words “greatest” and “highest” refer to the largest value, while “divisor” and “factor” refer to a number that divides another number exactly. So, whether you hear GCD, GCF, or HCF, they are all talking about the same concept. For example, if we take the numbers 8 and 12, the common factors are 1, 2, and 4. Out of these, 4 is the greatest one. So, the GCD (or HCF) of 8 and 12 is 4.
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.
Learn about Properties of Integers here.
The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder.
To find the GCD of given numbers, we commonly use the following theoretical methods:
This method involves expressing each number as a product of its prime factors. The GCD is then found by taking the product of the common prime factors raised to the lowest power for each.
Definition:
If two numbers are expressed as a product of their prime factors, then the GCD is the product of all common prime factors.
Example:
Find the GCD of 36 and 60.
So, GCD(36, 60) = 12
This is an efficient method to find the GCD of two numbers. It is based on the principle that the GCD of two numbers also divides their difference.
Definition:
If a and b are two natural numbers such that a > b, then GCD(a, b) = GCD(b, a mod b). The process is repeated until the remainder becomes zero. The last non-zero remainder is the GCD.
Example:
Find the GCD of 48 and 18.
So, GCD(48, 18) = 6
This method involves listing all the factors of the given numbers and identifying the greatest factor common to all.
Definition:
The GCD of two or more numbers is the largest number that appears in the list of factors common to all the numbers.
Example:
Find the GCD of 24 and 36.
The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), is a useful concept in many real-life situations. It helps us divide or arrange things in the most efficient and equal way. Let’s look at a few practical uses:
A shopkeeper has 360 balls and 150 bats to pack in a day. She wants to pack them in such a way that each set has the same number in a box, and they take up the least area of the box. What is the number that can be placed in each set for this packing purpose?
In the above problem, the greatest common divisor of 360 and 150 will be the required number.
Other applications like arranging students in rows and columns in equal number, dividing a group of people into smaller sections, etc.
Arranging Students: Suppose you have to arrange students in equal rows and columns for a school event. GCD helps in finding the maximum number of students per row or column without leaving anyone out.
Dividing Groups: If you want to divide a group of people into smaller equal teams, the GCD helps in finding the largest possible team size that evenly splits the group.
Cutting Material: When cutting long ropes or fabric rolls into equal pieces with no waste, GCD helps you find the longest piece length that can be used for all.
GCD is Always Less Than or Equal to Each Number
The GCD of two natural numbers is always less than or equal to both of them.
GCD of a Number and 0 is the Number Itself
The GCD of any number and 0 is the number itself.
For example, GCD(a, 0) = a.
GCD of Two Equal Numbers is the Number Itself
If both numbers are the same, then their GCD is the number itself.
Example: GCD(12, 12) = 12.
Commutative Property
The order of the numbers does not affect the GCD.
GCD(a, b) = GCD(b, a)
Associative Property
The GCD of three numbers follows the associative law.
GCD(a, GCD(b, c)) = GCD(GCD(a, b), c)
Relation Between GCD and LCM
The product of the GCD and LCM of two numbers is equal to the product of the numbers.
GCD(a, b) × LCM(a, b) = a × b
Multiplication by a Common Factor
If both numbers are multiplied by the same number, the GCD also gets multiplied by that number.
GCD(k × a, k × b) = k × GCD(a, b)
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 factors 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: Find 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: Find GCD of 560, 570, 265 using LCM Formula.
Solution:
Step1: Let’s calculate the GCD of the 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
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.
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