# Greatest common divisor and least common multiple

The greatest common divisor (GCF) and the least common multiple (LCM) are two values ​​that can be calculated from the divisors of two or more numbers.

Although both are calculated from the same information, the GCF and the LCM are interpreted very differently.

On the one hand, the GCF is the largest number by which two or more numbers can be divided. This, without leaving any residue.

Instead, the LCM is the smallest figure that satisfies the condition of being a multiple of all the elements of a set of numbers.

It should be specified that a number is a multiple of another when it contains it exactly n times. That is, a number b is a multiple of a when b = a * s , where s is an integer.

To better understand the difference we can use an example with the following numbers: 450, 765 and 135.

First, we disaggregate each figure into divisors. These are those numbers in which it is contained in another exactly an amount n of times.

450 = (3 ^ 2) * (5 ^ 2) * 2

765 = (3 ^ 2) * 5 * 17

135 = (3 ^ 3) * 5

So, to calculate the GCF we would take the common divisors to their lowest power:

GCF = (3 ^ 2) * 5 = 45

Likewise, for the lcm we would take all the dividers, even those that do not repeat themselves, raising them to their maximum power:

lcm = (3 ^ 3) * (5 ^ 2) * 2 * 17 = 22,950

## Relationship between GCF and LCM

When you have two numbers, the following formula is true:

That is, for 4,368 and 308

4,368 = (2 ^ 4) * 13 * 7 * 3

308 = (2 ^ 2) * 11 * 7

So, the lcm would be: (2 ^ 2) * 7 = 28

Therefore, the GCF could be solved in the formula:

GCD = 4.368 * 308/28 = 48.048

## Some properties

Some properties to consider are also:

• If we have two prime numbers (which can only be divided by themselves and one to obtain a whole number) the LCM is the total of their multiplication. Likewise, its greatest common factor is 1. For example, if we have 11 and 103, its LCM is 1133 and its GCF is 1.
• The greatest common divisor of two or more numbers is a divisor of the least common multiple of such numbers. This is because the calculation is made based on the same factors. For example, if we have 132, 336 and 1,314

132 = (2 ^ 2) * 3 * 11

336 = (2 ^ 4) * 3 * 7

1,314 = (3 ^ 2) * 73 * 2

Then,

GCF = 3 * 2 = 6

lcm = (2 ^ 4) * (3 ^ 2) * 7 * 11 * 73 = 809.424

And we verify that the LCM is a multiple of the GCF: 809.424 / 6 = 134.904

Greatest common divisor (GCF)

• Numeric sets
• Maximum (math)
• Law of the big numbers