The least common multiple (LCM) is a mathematical indicator that a student needs to know in order to work effectively with fractions. NOC is studied as part of the secondary school curriculum, and, despite the apparent complexity of the material, this topic will not cause problems for a student who knows the multiplication table and knows how to work with degrees.
LCM definition
Before starting to get acquainted with the LCM, it is necessary to understand its broader concept - we are talking about the definition of the term "common multiple" and its role in practical calculations.
A common multiple of several numbers is a natural number that can be divided by each of these numbers without a remainder. In other words, a common multiple of a series of integers is any integer that is divisible by each of the numbers in the given series.
In our case, we'll focus on common multiples of integers, none of which equals zero.
As for the number of natural numbers, in relation to which we can apply the concept of "common multiple", then there can be two, three, four or more of them in a series.
The most popular of the common multiples is the least common multiple - the LCM is the positive value of the smallest common multiple of all the numbers in the series.
NOC Examples
From the definition of the least common multiple and its mathematical essence, it follows that several numbers always have an LCM.
The shortest form for the least common multiple is:
- a1, a2, ..., ak of the form LCM (a1, a2, ..., ak).
In addition, in some sources you can find the following form of writing:
- a1, a2, ..., ak of the form [a1, a2, ..., ak].
To demonstrate an example, let's take the LCM of two integers: 4 and 5. The resulting expression will look like this:
- LCM(4, 5) = 20.
If we take the LCM for the following four numbers: 3, −9, 5, −15, we get the notation:
- LCM(3, −9, 5, −15) = 45.
Even the simplest writing examples show that finding the least common multiple for a group of numbers is far from easy, and the process of finding it can be quite complicated. There are special algorithms and techniques that are actively used when calculating the least common multiple.
How LCM and GCD are related
A value known in mathematical calculations, called the least common divisor (hereinafter referred to as GCD), is associated with LCM through the following theorem: “the least common multiple (LCM) of two positive integers a and b is equal to the product of numbers a and b divided by to the greatest common divisor (gcd) of a and b".
You can describe this theorem using a mathematical expression as follows:
- LCD (a, b) = a ⋅ b / GCD (a, b).
As a proof of this theorem, we present some mathematical research.
Let's say m is a certain multiple of a and b. Accordingly, m is divisible by a, and, by the definition of divisibility, there is some integer k, with which we can write the equality:
- m = a ⋅ k.
But, we also know that m is also divisible by b, so a ⋅ k is also divisible by b.
We will use the symbol d to denote the expression GCD (a, b). So we can write equality using expressions:
- a = a1 ⋅ d,
- b = b1 ⋅ d.
Here:
- a1 = a / d,
- b1 = b / d,
where a1 and b1 are relatively prime numbers.
The condition obtained above that a ⋅ k is divisible by b allows us to write the following expression: a1 ⋅ d ⋅ k is divisible by b1 ⋅ d, and this, in accordance with the properties of divisibility, is equivalent to the condition that a1 ⋅ k is divisible by b1 .
Therefore, according to the properties of coprime numbers, since a1 ⋅ k is divisible by b1, and a1 is not divisible by b1 (a1 and b1 are coprime numbers), then k must be divisible by b1. In this case, we must have some integer t for which the expression is true:
- k = b1 ⋅ t,
and since
- b1 = b / d,
then:
- k = b / d ⋅ t.
Substituting into the expression
- m = a ⋅ k
instead of k its expression is b / d ⋅ t, we arrive at the final equality:
- m = a ⋅ b / d ⋅ t.
So we got an equality that specifies the form of all common multiples of a and b. Since a and b are positive numbers by the condition, then for t = 1 we get their least positive common multiple, which is equal to a ⋅ b / d.
Thus, we have proved that
- LCD (a, b) = a ⋅ b / GCD (a, b).
Knowing the basic provisions and rules associated with LCM helps to better understand its practical significance in mathematics, and also allows you to actively use it as an applied unit in calculations in which knowledge of the LCM value is a prerequisite.
