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Greatest Common Divisor & Least Common Multiple

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Knowledge

What is the greatest common divisor (GCD)?

The divisor shared by several numbers is called the common divisor of these numbers, and the largest one is called the largest common divisor of these numbers.

For example: what is the GSD of 54 and 24.

The divisors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54.

The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

The common divisors of 54 and 24 are: 1, 2, 3, 6.

So the GSD of 54 and 24 is 6.

What is the least common multiple (LCM)?

In two or more natural numbers, if they have the same multiples, these multiples are their common multiples. The smallest multiple is called the least common multiple of these numbers.

For example: what is the LCM of 4 and 6.

The multiples of 4 are:4, 8, 12, 16, 24, 28, 32, 36, 40, 44, 48...

The multiples of 6 are:6, 12, 18, 24, 30, 36, 42, 48, 54...

The common multiples of 4 and 6 are: 12, 24, 36, 48.

So the LCM of 4 and 6 is 12.

what's the difference between GCD and LCM?

The biggest difference between the GCF and the LCM is that one is based upon what can divide evenly into two numbers (GCF), while the other depends on what number shared between two integers can be divided by the two integers (LCM). One must also consider if the numbers only share itself and 1 as common multiples of factors, than those numbers are not related to each other. That’s exactly what the GCF and LCM finds how two whole numbers relate to each other.

How to find the greatest common divisor (GCD)?

The main method to find the greatest common divisor of two integers are:

Two-sided method: List all divisors of two integers separately, and find the greatest common divisor.

Prime factor decomposition: List the prime factor decomposition formulas of two numbers respectively, and calculate the product of common terms.

Short division: divide two numbers by their common prime factor until the two numbers are mutually prime, the product of all divisors is the greatest common divisor.

You may ask how to find the greatest common divisor for multiple numbers. We strongly recommend that you use the greatest common divisor calculator above. You just need to enter the number to get the answer immediately, which is simple and fast. Of course, if you want to work out the answer by yourself, it's also great. Let's learn from the following example.

For example, find the GCD for numbers 18, 24, and 36.

Step 1: decompose the number 18, 24, and 36 into factors respectively.

Wait for a minute. Are you thinking "what is decompose numbers into factors" ? Ok, let's take a quick look. Every number is the product of a unique set of prime factors, a group of prime numbers (including repeats) that, when multiplied together, equals that number. You can find those prime factors for a given number, by using a process called decomposition. Such as 8 = 2 × 2 × 2 or 39 = 3 × 13.

So, in this question,

18 = 2 × 9 = 2 × 3 × 3

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3

Step 2: find their common factors: 2 and 3.

Step 3: multiply 2 and 3, then it gives us the GSC=6.

These three numbers are divided by 6 without a remainder:

18 : 6 = 3

24 : 6 = 4

36 : 6 = 6

How to find the least common multiple (LCM)?

There are two normal ways to find the least common multiple.

The first method: write out the first multiples of two numbers, and then choose among those multiples a number that is common to both numbers and is small.

The second method: decomposed the numbers into prime factors, then write out the multipliers included in the first expansion, and add the multipliers missing in the second expansion. Multiply the resulting multipliers to obtain LCM.

Example 1

What is the greatest common factor of 54 and 24?

The number 54 can be expressed as a product of two integers in several different ways:

54 × 1 = 27 × 2 = 18 × 3 = 9 × 6

Thus the complete list of divisors of 54 is (1,2,3,6,9,18,27,54). Similarly, the divisors of 24 are (1,2,3,4,6,8,12,24). The numbers that these two lists have in common are the common divisors of 54 and 24, that is (1,2,3,6).

Of these, the greatest is 6, so it is the greatest common divisor:

GCD (54, 24) = 6

Example 2

Find the LCM of 4 and 6

Multiples of 4 are:

4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...

Multiples of 6 are:

6,12,18,24,30,36,42,48,54,60,66,72,...

Common multiples of 4 and 6 are the numbers that are in both lists:

12,24,36,48,60,72,...

Among this list, the smallest number is 12, hence the least common multiple is 12.
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