Divisors and multiples. Least Common Multiple (LCM) – Definition, Examples and Properties

Let's consider the solution next task. The boy's step is 75 cm, and the girl's step is 60 cm. It is necessary to find the smallest distance at which they both take an integer number of steps.

Solution. The entire path that the guys will go through must be divisible by 60 and 70, since they must each take an integer number of steps. In other words, the answer must be a multiple of both 75 and 60.

First, we will write down all the multiples of the number 75. We get:

  • 75, 150, 225, 300, 375, 450, 525, 600, 675, … .

Now let's write down the numbers that will be multiples of 60. We get:

  • 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, … .

Now we find the numbers that are in both rows.

  • Common multiples of numbers would be 300, 600, etc.

The smallest of them is the number 300. In this case, it will be called the least common multiple of the numbers 75 and 60.

Returning to the condition of the problem, the smallest distance at which the guys will take an integer number of steps will be 300 cm. The boy will cover this path in 4 steps, and the girl will need to take 5 steps.

Determining Least Common Multiple

  • The least common multiple of two natural numbers a and b is the smallest natural number, which is a multiple of both a and b.

In order to find the least common multiple of two numbers, it is not necessary to write down all the multiples of these numbers in a row.

You can use the following method.

How to find the least common multiple

First you need to decompose these numbers into prime factors.

  • 60 = 2*2*3*5,
  • 75=3*5*5.

Now let’s write down all the factors that are in the expansion of the first number (2,2,3,5) and add to it all the missing factors from the expansion of the second number (5).

As a result, we get a series of prime numbers: 2,2,3,5,5. The product of these numbers will be the least common factor for these numbers. 2*2*3*5*5 = 300.

General scheme for finding the least common multiple

  • 1. Divide numbers into prime factors.
  • 2. Write down the prime factors that are part of one of them.
  • 3. Add to these factors all those that are in the expansion of the others, but not in the selected one.
  • 4. Find the product of all the written factors.

This method is universal. It can be used to find the least common multiple of any number of natural numbers.

How to find LCM (least common multiple)

A common multiple of two integers is an integer that is evenly divisible by both given numbers without leaving a remainder.

The least common multiple of two integers is the smallest of all integers that is divisible by both given numbers without leaving a remainder.

Method 1. You can find the LCM, in turn, for each of the given numbers, writing out in ascending order all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.

Example for numbers 6 and 9.
We multiply the number 6, sequentially, by 1, 2, 3, 4, 5.
We get: 6, 12, 18 , 24, 30
We multiply the number 9, sequentially, by 1, 2, 3, 4, 5.
We get: 9, 18 , 27, 36, 45
As you can see, the LCM for numbers 6 and 9 will be equal to 18.

This method is convenient when both numbers are small and it is easy to multiply them by a sequence of integers. However, there are cases when you need to find the LCM for two-digit or three-digit numbers, and also when there are three or even more initial numbers.

Method 2. You can find the LCM by factoring the original numbers into prime factors.
After decomposition, it is necessary to cross out identical numbers from the resulting series of prime factors. The remaining numbers of the first number will be a multiplier for the second, and the remaining numbers of the second will be a multiplier for the first.

Example for numbers 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing down the multiples of these numbers in a row. To do this, let’s factor 75 and 60 into simple factors:
75 = 3 * 5 * 5, a
60 = 2 * 2 * 3 * 5 .
As you can see, factors 3 and 5 appear in both rows. We mentally “cross out” them.
Let us write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we are left with the number 5, and when decomposing the number 60, we are left with 2 * 2
This means that in order to determine the LCM for the numbers 75 and 60, we need to multiply the remaining numbers from the expansion of 75 (this is 5) by 60, and multiply the numbers remaining from the expansion of 60 (this is 2 * 2) by 75. That is, for ease of understanding , we say that we are multiplying “crosswise”.
75 * 2 * 2 = 300
60 * 5 = 300
This is how we found the LCM for the numbers 60 and 75. This is the number 300.

Example. Determine the LCM for the numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But first, as always, let’s factorize all the numbers
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
To correctly determine the LCM, we select the smallest of all numbers (this is the number 12) and sequentially go through its factors, crossing them out if in at least one of the other rows of numbers we encounter the same factor that has not yet been crossed out.

Step 1. We see that 2 * 2 occurs in all series of numbers. Let's cross them out.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

Step 2. In the prime factors of the number 12, only the number 3 remains. But it is present in the prime factors of the number 24. We cross out the number 3 from both rows, while no actions are expected for the number 16.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

As you can see, when decomposing the number 12, we “crossed out” all the numbers. This means that the finding of the LOC is completed. All that remains is to calculate its value.
For the number 12, take the remaining factors of the number 16 (next in ascending order)
12 * 2 * 2 = 48
This is the NOC

As you can see, in this case, finding the LCM was somewhat more difficult, but when you need to find it for three or more numbers, this method allows you to do it faster. However, both methods of finding the LCM are correct.

Schoolchildren are given a lot of tasks in mathematics. Among them, very often there are problems with the following formulation: there are two meanings. How to find the least common multiple of given numbers? It is necessary to be able to perform such tasks, since the acquired skills are used to work with fractions when different denominators. In this article we will look at how to find LOC and basic concepts.

Before finding the answer to the question of how to find LCM, you need to define the term multiple. Most often, the formulation of this concept sounds as follows: a multiple of a certain value A is a natural number that will be divisible by A without a remainder. So, for 4, the multiples will be 8, 12, 16, 20, and so on, to the required limit.

Moreover, the number of divisors for a specific value can be limited, but the multiples are infinitely many. There is also the same value for natural values. This is an indicator that is divided into them without a remainder. Having understood the concept of the smallest value for certain indicators, let's move on to how to find it.

Finding the NOC

The least multiple of two or more exponents is the smallest natural number that is entirely divisible by all specified numbers.

There are several ways to find such a value, consider the following methods:

  1. If the numbers are small, then write down on a line all those divisible by it. Keep doing this until you find something in common among them. In writing, they are denoted by the letter K. For example, for 4 and 3, the smallest multiple is 12.
  2. If these are large or you need to find a multiple of 3 or more values, then you should use another technique that involves decomposing numbers into prime factors. First, lay out the largest one listed, then all the others. Each of them has its own number of multipliers. As an example, let's decompose 20 (2*2*5) and 50 (5*5*2). For the smaller one, underline the factors and add them to the largest one. The result will be 100, which will be the least common multiple of the above numbers.
  3. When finding 3 numbers (16, 24 and 36) the principles are the same as for the other two. Let's expand each of them: 16 = 2*2*2*2, 24=2*2*2*3, 36=2*2*3*3. Only two twos from the expansion of the number 16 were not included in the expansion of the largest. We add them and get 144, which is the smallest result for the previously indicated numerical values.

Now we know what the general technique is for finding the smallest value for two, three or more values. However, there are also private methods, helping to search for NOC if the previous ones do not help.

How to find GCD and NOC.

Private methods of finding

As with any mathematical section, there are special cases of finding LCM that help in specific situations:

  • if one of the numbers is divisible by the others without a remainder, then the lowest multiple of these numbers is equal to it (the LCM of 60 and 15 is 15);
  • relatively prime numbers have no common prime factors. Their smallest value is equal to the product of these numbers. Thus, for the numbers 7 and 8 it will be 56;
  • the same rule works for other cases, including special ones, which can be read about in specialized literature. This should also include cases of decomposition of composite numbers, which are the topic of individual articles and even candidate dissertations.

Special cases are less common than standard examples. But thanks to them, you can learn to work with fractions of varying degrees of complexity. This is especially true for fractions, where there are unequal denominators.

Few examples

Let's look at a few examples that will help you understand the principle of finding the least multiple:

  1. Find the LOC (35; 40). We first decompose 35 = 5*7, then 40 = 5*8. Add 8 to the smallest number and get LOC 280.
  2. NOC (45; 54). We decompose each of them: 45 = 3*3*5 and 54 = 3*3*6. We add the number 6 to 45. We get an LCM equal to 270.
  3. Well, the last example. There are 5 and 4. There are no prime multiples of them, so the least common multiple in this case will be their product, equal to 20.

Thanks to the examples, you can understand how the NOC is located, what the nuances are and what the meaning of such manipulations is.

Finding NOC is much easier than it might initially seem. To do this, both simple expansion and multiplication are used simple values on top of each other. The ability to work with this section of mathematics helps with further study of mathematical topics, especially fractions varying degrees complexity.

Don't forget to solve examples periodically various methods, this develops the logical apparatus and allows you to remember numerous terms. Learn how to find such an exponent and you will be able to do well in the rest of the math sections. Happy learning math!

Video

This video will help you understand and remember how to find the least common multiple.

How to find the least common multiple?

    We need to find each factor of each of the two numbers for which we find the least common multiple, and then multiply by each other the factors that coincide in the first and second numbers. The result of the product will be the required multiple.

    For example, we have the numbers 3 and 5 and we need to find the LCM (least common multiple). Us need to multiply and three and five for all numbers starting from 1 2 3 ... and so on until we see the same number in both places.

    Multiply three and get: 3, 6, 9, 12, 15

    Multiply by five and get: 5, 10, 15

    The prime factorization method is the most classic method for finding the least common multiple (LCM) of several numbers. This method is clearly and simply demonstrated in the following video:

    Adding, multiplying, dividing, reducing to a common denominator and other arithmetic operations are very exciting activity, I especially admire the examples that take up an entire sheet.

    So find the common multiple of two numbers, which will be the smallest number by which the two numbers are divided. I would like to note that it is not necessary to resort to formulas in the future to find what you are looking for, if you can count in your head (and this can be trained), then the numbers themselves pop up in your head and then the fractions crack like nuts.

    First, let's learn that you can multiply two numbers by each other, and then reduce this number and divide alternately by these two numbers, so we will find the smallest multiple.

    For example, two numbers 15 and 6. Multiply and get 90. This is obvious larger number. Moreover, 15 is divisible by 3 and 6 is divisible by 3, which means we also divide 90 by 3. We get 30. We try 30 divide 15 equals 2. And 30 divide 6 equals 5. Since 2 is the limit, it turns out that the least multiple for numbers is 15 and 6 will be 30.

    With larger numbers it will be a little more difficult. but if you know which numbers give a zero remainder when dividing or multiplying, then, in principle, there are no great difficulties.

  • How to find NOC

    Here is a video that will give you two ways to find the least common multiple (LCM). After practicing using the first of the suggested methods, you can better understand what the least common multiple is.

    • I present another way to find the least common multiple. Let's look at it with a clear example.

    You need to find the LCM of three numbers at once: 16, 20 and 28.

    • We represent each number as a product of its prime factors:
    • We write down the powers of all prime factors:

    16 = 224 = 2^24^1

    20 = 225 = 2^25^1

    28 = 227 = 2^27^1

    • We select all prime divisors (multipliers) with the greatest powers, multiply them and find the LCM:

    LCM = 2^24^15^17^1 = 4457 = 560.

    LCM(16, 20, 28) = 560.

    Thus, the result of the calculation was the number 560. It is the least common multiple, that is, it is divisible by each of the three numbers without a remainder.

    The least common multiple is a number that can be divided into several given numbers without leaving a remainder. In order to calculate such a figure, you need to take each number and decompose it into simple factors. Those numbers that match are removed. Leaves everyone one at a time, multiply them among themselves in turn and get the desired one - the least common multiple.

    NOC, or least common multiple, is the smallest natural number of two or more numbers that is divisible by each of the given numbers without a remainder.

    Here is an example of how to find the least common multiple of 30 and 42.

    • The first step is to factor these numbers into prime factors.

    For 30 it is 2 x 3 x 5.

    For 42, this is 2 x 3 x 7. Since 2 and 3 are in the expansion of the number 30, we cross them out.

    • We write out the factors that are included in the expansion of the number 30. This is 2 x 3 x 5.
    • Now we need to multiply them by the missing factor, which we have when expanding 42, which is 7. We get 2 x 3 x 5 x 7.
    • We find what 2 x 3 x 5 x 7 is equal to and get 210.

    As a result, we find that the LCM of the numbers 30 and 42 is 210.

    To find the least common multiple, you need to perform several sequentially simple actions. Let's look at this using two numbers as an example: 8 and 12

    1. We factor both numbers into prime factors: 8=2*2*2 and 12=3*2*2
    2. We reduce the same factors of one of the numbers. In our case, 2 * 2 coincide, let’s reduce them for the number 12, then 12 will have one factor left: 3.
    3. Find the product of all remaining factors: 2*2*2*3=24

    Checking, we make sure that 24 is divisible by both 8 and 12, and this is the smallest natural number that is divisible by each of these numbers. Here we are found the least common multiple.

    I’ll try to explain using the numbers 6 and 8 as an example. The least common multiple is a number that can be divided by these numbers (in our case, 6 and 8) and there will be no remainder.

    So, we first start multiplying 6 by 1, 2, 3, etc. and 8 by 1, 2, 3, etc.

The least common multiple of two numbers is directly related to the greatest common divisor of those numbers. This connection between GCD and NOC is determined by the following theorem.

Theorem.

The least common multiple of two positive integers a and b is equal to the product of a and b divided by the largest common divisor numbers a and b, that is, LCM(a, b)=a b:GCD(a, b).

Proof.

Let M is some multiple of the numbers a and b. That is, M is divisible by a, and by the definition of divisibility, there is some integer k such that the equality M=a·k is true. But M is also divisible by b, then a·k is divisible by b.

Let's denote gcd(a, b) as d. Then we can write the equalities a=a 1 ·d and b=b 1 ·d, and a 1 =a:d and b 1 =b:d will be relatively prime numbers. Consequently, the condition obtained in the previous paragraph that a · k is divisible by b can be reformulated as follows: a 1 · d · k is divided by b 1 · d , and this, due to divisibility properties, is equivalent to the condition that a 1 · k is divisible by b 1.

You also need to write down two important corollaries from the theorem considered.

    The common multiples of two numbers are the same as the multiples of their least common multiple.

    This is indeed the case, since any common multiple of M of the numbers a and b is determined by the equality M=LMK(a, b)·t for some integer value t.

    Least common multiple of coprime positive numbers a and b are equal to their product.

    The rationale for this fact is quite obvious. Since a and b are relatively prime, then gcd(a, b)=1, therefore, GCD(a, b)=a b: GCD(a, b)=a b:1=a b.

Least common multiple of three or more numbers

Finding the least common multiple of three or more numbers can be reduced to sequentially finding the LCM of two numbers. How this is done is indicated in the following theorem. a 1 , a 2 , …, a k coincide with the common multiples of the numbers m k-1 and a k , therefore, coincide with the common multiples of the number m k . And since the smallest positive multiple of the number m k is the number m k itself, then the smallest common multiple of the numbers a 1, a 2, ..., a k is m k.

References.

  • Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
  • Vinogradov I.M. Fundamentals of number theory.
  • Mikhelovich Sh.H. Number theory.
  • Kulikov L.Ya. and others. Collection of problems in algebra and number theory: Tutorial for students of physics and mathematics. specialties of pedagogical institutes.


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