What is the least common multiple of numbers. Divisors and multiples

Let's start studying the least common multiple of two or more numbers. In this section we will define the term, consider the theorem that establishes the connection between the least common multiple and the greatest common divisor, and give examples of solving problems.

Common multiples – definition, examples

In this topic, we will be interested only in common multiples of integers other than zero.

Definition 1

Common multiple of integers is an integer that is a multiple of all given numbers. In fact, it is any integer that can be divided by any of the given numbers.

The definition of common multiples refers to two, three, or more integers.

Example 1

According to the definition given above, the common multiples of the number 12 are 3 and 2. Also, the number 12 will be a common multiple of the numbers 2, 3 and 4. The numbers 12 and -12 are common multiples of the numbers ±1, ±2, ±3, ±4, ±6, ±12.

At the same time, the common multiple of numbers 2 and 3 will be the numbers 12, 6, − 24, 72, 468, − 100,010,004 and a whole series of others.

If we take numbers that are divisible by the first number of a pair and not divisible by the second, then such numbers will not be common multiples. So, for numbers 2 and 3, the numbers 16, − 27, 5009, 27001 will not be common multiples.

0 is a common multiple of any set of integers other than zero.

If we recall the property of divisibility with respect to opposite numbers, it turns out that some integer k will be a common multiple of these numbers, just like the number - k. This means that common divisors can be either positive or negative.

Is it possible to find the LCM for all numbers?

The common multiple can be found for any integer.

Example 2

Suppose we are given k integers a 1 , a 2 , … , a k. The number we get when multiplying numbers a 1 · a 2 · … · a k according to the property of divisibility, it will be divided into each of the factors that were included in the original product. This means that the product of numbers a 1 , a 2 , … , a k is the least common multiple of these numbers.

How many common multiples can these integers have?

A group of integers can have a large number of common multiples. In fact, their number is infinite.

Example 3

Suppose we have some number k. Then the product of the numbers k · z, where z is an integer, will be a common multiple of the numbers k and z. Given that the number of numbers is infinite, the number of common multiples is infinite.

Least Common Multiple (LCM) – Definition, Notation and Examples

Recall the concept of the smallest number from a given set of numbers, which we discussed in the section “Comparing Integers.” Taking this concept into account, we formulate the definition of the least common multiple, which has the greatest practical significance among all common multiples.

Definition 2

Least common multiple of given integers is the smallest positive common multiple of these numbers.

A least common multiple exists for any number of given numbers. The most commonly used abbreviation for the concept in reference literature is NOC. Short notation for least common multiple of numbers a 1 , a 2 , … , a k will have the form LOC (a 1 , a 2 , … , a k).

Example 4

The least common multiple of 6 and 7 is 42. Those. LCM(6, 7) = 42. The least common multiple of the four numbers 2, 12, 15 and 3 is 60. A short notation will look like LCM (- 2, 12, 15, 3) = 60.

The least common multiple is not obvious for all groups of given numbers. Often it has to be calculated.

Relationship between NOC and GCD

The least common multiple and the greatest common divisor are related. The relationship between concepts is established by the theorem.

Theorem 1

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

Evidence 1

Suppose we have some number M, which is a multiple of the numbers a and b. If the number M is divisible by a, there is also some integer z , under which the equality is true M = a k. According to the definition of divisibility, if M is divisible by b, so then a · k divided by b.

If we introduce a new notation for gcd (a, b) as d, then we can use the equalities a = a 1 d and b = b 1 · d. In this case, both equalities will be relatively prime numbers.

We have already established above that a · k divided by b. Now this condition can be written as follows:
a 1 d k divided by b 1 d, which is equivalent to the condition a 1 k divided by b 1 according to the properties of divisibility.

According to the property of coprime numbers, if a 1 And b 1– coprime numbers, a 1 not divisible by b 1 despite the fact that a 1 k divided by b 1, That b 1 must be shared k.

In this case, it would be appropriate to assume that there is a number t, for which k = b 1 t, and since b 1 = b: d, That k = b: d t.

Now instead of k let's substitute into equality M = a k expression of the form b: d t. This allows us to achieve equality M = a b: d t. At t = 1 we can get the least positive common multiple of a and b , equal a b: d, provided that numbers a and b positive.

So we proved that LCM (a, b) = a · b: GCD (a, b).

Establishing a connection between LCM and GCD allows you to find the least common multiple through the greatest common divisor of two or more given numbers.

Definition 3

The theorem has two important consequences:

• multiples of the least common multiple of two numbers are the same as the common multiples of those two numbers;
• the least common multiple of mutually prime positive numbers a and b is equal to their product.

It is not difficult to substantiate these two facts. Any common multiple of M of numbers a and b is defined by the equality M = LCM (a, b) · t for some integer value t. 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

In order to find the least common multiple of several numbers, it is necessary to sequentially find the LCM of two numbers.

Theorem 2

Let's pretend that a 1 , a 2 , … , a k are some positive integers. In order to calculate the LCM m k these numbers, we need to sequentially calculate m 2 = LCM(a 1 , a 2) , m 3 = NOC(m 2 , a 3) , … , m k = NOC(m k - 1 , a k) .

Evidence 2

The first corollary from the first theorem discussed in this topic will help us prove the validity of the second theorem. The reasoning is based on the following algorithm:

• common multiples of numbers a 1 And a 2 coincide with multiples of their LCM, in fact, they coincide with multiples of the number m 2;
• common multiples of numbers a 1, a 2 And a 3 m 2 And a 3 m 3;
• common multiples of numbers a 1 , a 2 , … , a k coincide with common multiples of numbers m k - 1 And a k, therefore, coincide with multiples of the number m k;
• due to the fact that the smallest positive multiple of the number m k is the number itself m k, then the least common multiple of the numbers a 1 , a 2 , … , a k is m k.

This is how we proved the theorem.

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Divisibility criteria for natural numbers.

Numbers divisible by 2 without a remainder are calledeven .

Numbers that are not evenly divisible by 2 are calledodd .

Test for divisibility by 2

If a natural number ends with an even digit, then this number is divisible by 2 without a remainder, and if a number ends with an odd digit, then this number is not evenly divisible by 2.

For example, the numbers 60 , 30 8 , 8 4 are divisible by 2 without remainder, and the numbers are 51 , 8 5 , 16 7 are not divisible by 2 without a remainder.

Test for divisibility by 3

If the sum of the digits of a number is divisible by 3, then the number is divisible by 3; If the sum of the digits of a number is not divisible by 3, then the number is not divisible by 3.

For example, let’s find out whether the number 2772825 is divisible by 3. To do this, let’s calculate the sum of the digits of this number: 2+7+7+2+8+2+5 = 33 - divisible by 3. This means the number 2772825 is divisible by 3.

Divisibility test by 5

If the record of a natural number ends with the digit 0 or 5, then this number is divisible by 5 without a remainder. If the record of a number ends with another digit, then the number is not divisible by 5 without a remainder.

For example, the numbers 15 , 3 0 , 176 5 , 47530 0 are divisible by 5 without remainder, and the numbers are 17 , 37 8 , 9 1 don't share.

Divisibility test by 9

If the sum of the digits of a number is divisible by 9, then the number is divisible by 9; If the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9.

For example, let’s find out whether the number 5402070 is divisible by 9. To do this, let’s calculate the sum of the digits of this number: 5+4+0+2+0+7+0 = 16 - not divisible by 9. This means the number 5402070 is not divisible by 9.

Divisibility test by 10

If a natural number ends with the digit 0, then this number is divisible by 10 without a remainder. If a natural number ends with another digit, then it is not evenly divisible by 10.

For example, the numbers 40 , 17 0 , 1409 0 are divisible by 10 without remainder, and the numbers 17 , 9 3 , 1430 7 - don't share.

The rule for finding the greatest common divisor (GCD).

To find the greatest common divisor of several natural numbers, you need to:

2) from the factors included in the expansion of one of these numbers, cross out those that are not included in the expansion of other numbers;

3) find the product of the remaining factors.

Example. Let's find GCD (48;36). Let's use the rule.

1. Let's factor the numbers 48 and 36 into prime factors.

48 = 2 · 2 · 2 · 2 · 3

36 = 2 · 2 · 3 · 3

2. From the factors included in the expansion of the number 48, we delete those that are not included in the expansion of the number 36.

48 = 2 · 2 · 2 · 2 · 3

The remaining factors are 2, 2 and 3.

3. Multiply the remaining factors and get 12. This number is the greatest common divisor of the numbers 48 and 36.

GCD (48;36) = 2· 2 · 3 = 12.

The rule for finding the least common multiple (LCM).

To find the least common multiple of several natural numbers, you need to:

1) factor them into prime factors;

2) write down the factors included in the expansion of one of the numbers;

3) add to them the missing factors from the expansions of the remaining numbers;

4) find the product of the resulting factors.

Example. Let's find the LOC (75;60). Let's use the rule.

1. Let's factor the numbers 75 and 60 into prime factors.

75 = 3 · 5 · 5

60 = 2 · 2 · 3 · 3

2. Let’s write down the factors included in the expansion of the number 75: 3, 5, 5.

LCM(75;60) = 3 · 5 · 5 · …

3. Add to them the missing factors from the expansion of the number 60, i.e. 2, 2.

LCM(75;60) = 3 · 5 · 5 · 2 · 2

4. Find the product of the resulting factors

LCM(75;60) = 3 · 5 · 5 · 2 · 2 = 300.

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 with 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 like this: 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.

In this case, 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 master's theses.

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.

Some 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 the 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, which is 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 of simple values ​​by each other are used. The ability to work with this section of mathematics helps with further study of mathematical topics, especially fractions of varying degrees of complexity.

Don’t forget to periodically solve examples using different methods; this develops your 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.

Definition. The largest natural number that can be divided without a remainder by numbers a and b is called greatest common divisor (GCD) these numbers.

Let's find the greatest common divisor of the numbers 24 and 35.
The divisors of 24 are the numbers 1, 2, 3, 4, 6, 8, 12, 24, and the divisors of 35 are the numbers 1, 5, 7, 35.
We see that the numbers 24 and 35 have only one common divisor - the number 1. Such numbers are called mutually prime.

Definition. Natural numbers are called mutually prime, if their greatest common divisor (GCD) is 1.

Greatest Common Divisor (GCD) can be found without writing out all the divisors of the given numbers.

Let's factor the numbers 48 and 36 and get:
48 = 2 * 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3.
From the factors included in the expansion of the first of these numbers, we cross out those that are not included in the expansion of the second number (i.e., two twos).
The factors remaining are 2 * 2 * 3. Their product is equal to 12. This number is the greatest common divisor of the numbers 48 and 36. The greatest common divisor of three or more numbers is also found.

To find greatest common divisor

2) from the factors included in the expansion of one of these numbers, cross out those that are not included in the expansion of other numbers;
3) find the product of the remaining factors.

If all given numbers are divisible by one of them, then this number is greatest common divisor given numbers.
For example, the greatest common divisor of the numbers 15, 45, 75 and 180 is the number 15, since all other numbers are divisible by it: 45, 75 and 180.

Least common multiple (LCM)

Definition. Least common multiple (LCM) natural numbers a and b is the smallest natural number that is a multiple of both a and b. The least common multiple (LCM) 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 prime factors: 75 = 3 * 5 * 5, and 60 = 2 * 2 * 3 * 5.
Let's write down the factors included in the expansion of the first of these numbers, and add to them the missing factors 2 and 2 from the expansion of the second number (i.e., we combine the factors).
We get five factors 2 * 2 * 3 * 5 * 5, the product of which is 300. This number is the least common multiple of the numbers 75 and 60.

They also find the least common multiple of three or more numbers.

To find least common multiple several natural numbers, you need:
1) factor them into prime factors;
2) write down the factors included in the expansion of one of the numbers;
3) add to them the missing factors from the expansions of the remaining numbers;
4) find the product of the resulting factors.

Note that if one of these numbers is divisible by all other numbers, then this number is the least common multiple of these numbers.
For example, the least common multiple of the numbers 12, 15, 20, and 60 is 60 because it is divisible by all of those numbers.

Pythagoras (VI century BC) and his students studied the question of the divisibility of numbers. They called a number equal to the sum of all its divisors (without the number itself) a perfect number. For example, the numbers 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The next perfect numbers are 496, 8128, 33,550,336. The Pythagoreans only knew the first three perfect numbers. The fourth - 8128 - became known in the 1st century. n. e. The fifth - 33,550,336 - was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there are odd perfect numbers or whether there is a largest perfect number.
The interest of ancient mathematicians in prime numbers is due to the fact that any number is either prime or can be represented as a product of prime numbers, i.e. prime numbers are like bricks from which the rest of the natural numbers are built.
You probably noticed that prime numbers in the series of natural numbers occur unevenly - in some parts of the series there are more of them, in others - less. But the further we move along the number series, the less common prime numbers are. The question arises: is there a last (largest) prime number? The ancient Greek mathematician Euclid (3rd century BC), in his book “Elements”, which was the main textbook of mathematics for two thousand years, proved that there are infinitely many prime numbers, i.e. behind every prime number there is an even greater prime number.
To find prime numbers, another Greek mathematician of the same time, Eratosthenes, came up with this method. He wrote down all the numbers from 1 to some number, and then crossed out one, which is neither a prime nor a composite number, then crossed out through one all the numbers coming after 2 (numbers that are multiples of 2, i.e. 4, 6 , 8, etc.). The first remaining number after 2 was 3. Then, after two, all numbers coming after 3 (numbers that were multiples of 3, i.e. 6, 9, 12, etc.) were crossed out. in the end only the prime numbers remained uncrossed.

The online calculator allows you to quickly find the greatest common divisor and least common multiple for two or any other number of numbers.

Calculator for finding GCD and LCM

Find GCD and LCM

Found GCD and LOC: 6433

How to use the calculator

• Enter numbers in the input field
• If you enter incorrect characters, the input field will be highlighted in red
• click the "Find GCD and LOC" button

How to enter numbers

• Numbers are entered separated by a space, period or comma
• The length of entered numbers is not limited, so finding GCD and LCM of long numbers is not difficult

What are GCD and NOC?

Greatest common divisor several numbers is the largest natural integer by which all original numbers are divisible without a remainder. The greatest common divisor is abbreviated as GCD.
Least common multiple several numbers is the smallest number that is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.

How to check that a number is divisible by another number without a remainder?

To find out whether one number is divisible by another without a remainder, you can use some properties of divisibility of numbers. Then, by combining them, you can check the divisibility of some of them and their combinations.

Some signs of divisibility of numbers

1. Divisibility test for a number by 2
To determine whether a number is divisible by two (whether it is even), it is enough to look at the last digit of this number: if it is equal to 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine whether the number 34938 is divisible by 2.
Solution: We look at the last digit: 8 - that means the number is divisible by two.

2. Divisibility test for a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine whether a number is divisible by 3, you need to calculate the sum of the digits and check whether it is divisible by 3. Even if the sum of the digits is very large, you can repeat the same process again.
Example: determine whether the number 34938 is divisible by 3.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 3, which means the number is divisible by three.

3. Divisibility test for a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine whether the number 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.

4. Divisibility test for a number by 9
This sign is very similar to the sign of divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine whether the number 34938 is divisible by 9.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 9, which means the number is divisible by nine.

How to find GCD and LCM of two numbers

How to find the gcd of two numbers

The easiest way to calculate the greatest common divisor of two numbers is to find all possible divisors of those numbers and choose the largest one.

Let's consider this method using the example of finding GCD(28, 36):

1. We factor both numbers: 28 = 1·2·2·7, 36 = 1·2·2·3·3
2. We find common factors, that is, those that both numbers have: 1, 2 and 2.
3. We calculate the product of these factors: 1 2 2 = 4 - this is the greatest common divisor of the numbers 28 and 36.

How to find the LCM of two numbers

There are two most common ways to find the least multiple of two numbers. The first method is that you can write down the first multiples of two numbers, and then choose among them a number that will be common to both numbers and at the same time the smallest. And the second is to find the gcd of these numbers. Let's consider only it.

To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. Let's find the LCM for the same numbers 28 and 36:

1. Find the product of numbers 28 and 36: 28·36 = 1008
2. GCD(28, 36), as already known, is equal to 4
3. LCM(28, 36) = 1008 / 4 = 252 .

Finding GCD and LCM for several numbers

The greatest common divisor can be found for several numbers, not just two. To do this, the numbers to be found for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. You can also use the following relation to find the gcd of several numbers: GCD(a, b, c) = GCD(GCD(a, b), c).

A similar relationship applies to the least common multiple: LCM(a, b, c) = LCM(LCM(a, b), c)

Example: find GCD and LCM for numbers 12, 32 and 36.

1. First, let's factorize the numbers: 12 = 1·2·2·3, 32 = 1·2·2·2·2·2, 36 = 1·2·2·3·3.
2. Let's find the common factors: 1, 2 and 2.
3. Their product will give GCD: 1·2·2 = 4
4. Now let’s find the LCM: to do this, let’s first find the LCM(12, 32): 12·32 / 4 = 96 .
5. To find the LCM of all three numbers, you need to find GCD(96, 36): 96 = 1·2·2·2·2·2·3 , 36 = 1·2·2·3·3 , GCD = 1·2· 2 3 = 12.
6. LCM(12, 32, 36) = 96·36 / 12 = 288.