how to find the greatest common factor

Greatest Mutual Divisor - GCD

The greatest mutual divisor (GCD) refers to the greatest positive integer that is a common divisor for a given gear up of positive integers. It is likewise termed as the highest mutual gene (HCF) or the greatest common factor (GCF). In this lesson, we will learn how to find the greatest mutual divisor in detail.

1.	What is Greatest Common Divisor?
2.	How to Find the Greatest Mutual Divisor?
3.	Finding Greatest Mutual Divisor past LCM Method
4.	Euclid's Algorithm for Greatest Common Divisor
5.	FAQs on Greatest Mutual Divisor

What is Greatest Common Divisor?

For a fix of positive integers (a, b), the greatest common divisor is defined as the greatest positive number which is a common gene of both the positive integers (a, b). GCD of any ii numbers is never negative or 0 as the least positive integer common to whatever 2 numbers is always ane. At that place are 2 ways to make up one's mind the greatest common divisor of ii numbers:

Past finding the mutual divisors
Past Euclid's algorithm

How to Find the Greatest Common Divisor?

For a fix of 2 positive integers (a, b) we utilise the beneath-given steps to find the greatest common divisor:

Step i: Write the divisors of positive integer "a".
Step 2: Write the divisors of positive integer "b".
Footstep iii: Enlist the common divisors of "a" and "b".
Pace 4: Now find the divisor which is the highest of both "a" and "b".

Example: Observe the greatest common divisor of thirteen and 48.
Solution: We will employ the below steps to determine the greatest mutual divisor of (thirteen, 48).

Divisors of 13 are ane, and xiii.
Divisors of 48 are i, 2, three, four, vi, viii, 12, 16, 24 and 48.

The mutual divisor of 13 and 48 is i.
The greatest common divisor of 13 and 48 is 1.

Thus, GCD(13, 48) = 1.

Finding Greatest Common Divisor by LCM Method

Equally per the LCM Method for the greatest common divisor, the GCD of two positive integers (a, b) can be calculated past using the following formula:

Greatest Common Divisor using LCM Method

The steps to calculate the GCD of (a, b) using the LCM method is:

Stride 1: Observe the product of a and b.
Pace 2: Find the least mutual multiple (LCM) of a and b.
Stride 3: Divide the values obtained in Stride ane and Step two.
Step 4: The obtained value later sectionalization is the greatest common divisor of (a, b).

Instance: Find the greatest mutual divisor of 15 and 70 using the LCM method.
Solution: The greatest common divisor of 15 and lxx can be calculated as:

The product of xv and 70 is given as, 15 × 70
The LCM of (15, 70) is 210.
GCD (15, xx) = (15 × lxx)/ 210 = five.

∴ The greatest common divisor of (15, 70) is 5.

Euclid'southward Algorithm for Greatest Common Divisor

Equally per Euclid's algorithm for the greatest mutual divisor, the GCD of two positive integers (a, b) can be calculated every bit:

If a = 0, then GCD (a, b) = b as GCD (0, b) = b.
If b = 0, so GCD (a, b) = a as GCD (a, 0) = a.
If both a≠0 and b≠0, nosotros write 'a' in quotient rest form (a = b×q + r) where q is the quotient and r is the remainder, and a>b.
Observe the GCD (b, r) as GCD (b, r) = GCD (a, b)
We echo this process until we get the remainder as 0.

Instance: Find the GCD of 12 and ten using Euclid's Algorithm.
Solution: The GCD of 12 and ten tin be institute using the below steps:
a = 12 and b = 10
a≠0 and b≠0
In quotient remainder form we can write 12 = ten × 1 + two
Thus, GCD (ten, 2) is to exist found, as GCD(12, 10) = GCD (ten, 2)

Now, a = 10 and b = 2
a≠0 and b≠0
In quotient remainder form we can write 10 = 2 × five + 0
Thus, GCD (two,0) is to exist found, as GCD(10, 2) = GCD (2, 0)

Now, a = 2 and b = 0
a≠0 and b=0
Thus, GCD (2,0) = two

GCD (12, 10) = GCD (x, 2) = GCD (ii, 0) = 2

Thus, GCD of 12 and 10 is 2.

Euclid'south algorithm is very useful to discover GCD of larger numbers, equally in this nosotros tin can easily break down numbers into smaller numbers to find the greatest mutual divisor.

Topics Related to Greatest Common Divisor

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Examples on Greatest Common Divisor

Example 1: Decide the greatest common divisor of 12 and 26.

Solution: The greatest common divisor of 12 and 26 can be calculated as:

Divisors of 12 are 1, two, 3, 4, 6, and 12.
Divisors of 26 are 1, two, 13, and 26.

Common divisors of 12 and 26 are ane and 2.
∴ The greatest mutual divisor of (12, 26) is ii.
Case 2: Using the LCM method, decide the value of the greatest common divisor of 20 and 65.

Solution: We tin can summate the value of the greatest common divisor of 20 and 65 using the following steps:
- The product of xx and 65 is given as, 20 × 65
- The LCM of (xx, 65) is 260.
- GCD (20, 65) = (20 × 65)/ 260 = v.
∴ The greatest common divisor of (20, 65) is v.

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Greatest Mutual Divisor Exercise Questions

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FAQs on Greatest Mutual Divisor

What does Greatest Common Divisor Mean?

The greatest common divisor for any two positive integers (a, b) is the greatest factor which is common to both the integers a and b. It is also known as the highest common factor or greatest mutual cistron.

How do you find the Greatest Common Divisor of 2 Numbers?

The greatest mutual divisor of two numbers can exist determined using the following steps:

Stride 1: Find the divisors of positive integer "a".
Step two: Observe the divisors of positive integer "b".
Step 3: Lis the divisors common to "a" and "b".
Step 4: Observe the divisor which is the highest of all the common divisors of both "a" and "b".

What is LCM Method for Greatest Common Divisor?

We tin can decide the value of the greatest common divisor past using the LCM method. As per the LCM method, we can obtain the GCD of any two positive integers by finding the product of both the numbers and the least common multiple of both numbers. LCM method to obtain the greatest common divisor is given as GCD (a, b) = (a × b)/ LCM (a, b).

How to Observe the Greatest Mutual Divisor Using LCM Method?

Nosotros can find the GCD of (a, b) using the LCM method by using the following steps:

Step i: Determine the product of a and b.
Step 2: At present, find the to the lowest degree common multiple (LCM) of a and b.
Step 3: Divide the values obtained in Stride one and Footstep two.
Step 4: The obtained value after division is the greatest common divisor of (a, b).

Can the Greatest Mutual Divisor be Negative?

No, the greatest common divisor cannot exist negative equally it represents the greatest common divisor of 2 positive integers. The least value of GCD tin can be ane and non lesser than it. This proves the bespeak that GCD cannot concord a negative value.

Are GCD and HCF the Same?

Yep, GCD and HCF are the aforementioned. In either example, the value of GCD, HCF can be adamant past checking the common divisors or factors so finding the greatest divisor of both the numbers.

Source: https://www.cuemath.com/numbers/greatest-common-divisor-gcd/

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