What is the greatest common divisor (GCLN), how to find the greatest common divisor? Please refer to the article below to get the answer.

## Greatest common divisor

## What is the greatest common divisor?

The greatest common divisor (GCC) of two or more numbers is the largest number in the set of common divisors of those numbers.

In English, the greatest common divisor is called the greatest common factor (GCF).

The symbol for the greatest common divisor of a and b is GCLN(a,b).

Example: Find GCLN(24, 16, 32)

U(24) = {1, 2, 3, 4, 6, 8, 12, 24}

U(16) = {1, 2, 4, 8, 16}

U(32) = {1, 2, 4, 8, 16, 32}

So GCC(24, 16, 32) = 8

## How to find the greatest common divisor?

### Method 1: List the common divisors of the numbers and then choose the GCC

To find the greatest common divisor of numbers, we find the set of divisors of each of those numbers. Then choose the greatest common divisor.

Example: Find the greatest common divisor of two natural numbers 16 and 30.

First we find the set of divisors of 16 and 30.

U(15) = { 1, 2, 4 , 8, 16 }

U(30) = { 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 }

So GCC (16,30) = 2

### Method 2: Parsing numbers into prime factors

Step 1: Factor each number into prime factors.

Step 2: Pick out the common prime factors.

Step 3: Build the product of the selected factors, each factor taken with its smallest exponent.

That product is the GCLN to find.

Example: Find UCLN(12, 30)

12 = 2 x 2 x 3

30 = 2 x 3 x 5

We have: the common prime factors are 2 and 3.

So GCLN(12, 30) = 2 x 3 = 6

### Method 3: Find the LCC by Least Common Multiple (BCNN) (conditions a, b are not 0)

The greatest common divisor of a and b can be calculated by dividing the product of a and b by the least common multiple (BCNN) of a and b.

Example: Find CCLN(12, 30)

B(12) = {0, 12, 24, 36, 48, 60,…}

B(30) = {0, 30, 60,…}

We have: BCNN(12,30) = 60

So GCC(12,30) = 12.30:60 = 6

## Tips for finding the greatest common divisor

- If one of the given numbers is 1, then the greatest common divisor of those numbers is 1.

For example: UCLN(1, 55, 95) = 1

- If the given numbers have no common prime factors, then the greatest common divisor of that number is 1.

For example: Numbers 5 and 8 have no common prime factors, so GCLN(5,8) = 1

- Two or more numbers whose greatest common divisor is 1 are called co-prime numbers.

For example, GCLN (6,35) = 1 so 6 and 35 are co-prime.

- Among the given numbers, if the smallest number is the divisor of the remaining numbers, then the greatest common divisor of the given numbers is that smallest number.

For example: 5 is both a divisor of 5 and 15, so GCLN(5,15) = 5

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