Finding the least common divisor. Least Common Multiple (LCM) - Definition, Examples and Properties

But many natural numbers are evenly divisible by other natural numbers.

For example:

The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;

The number 36 is divisible by 1, by 2, by 3, by 4, by 6, by 12, by 18, by 36.

The numbers by which the number is divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called number divisors. Divisor of a natural number a is the natural number that divides the given number a without a trace. A natural number that has more than two factors is called composite .

Note that the numbers 12 and 36 have common divisors. These are the numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. The common divisor of these two numbers a and b is the number by which both given numbers are divisible without a remainder a and b.

common multiple several numbers is called the number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all jcommon multiples, there is always the smallest one, in this case it is 90. This number is called leastcommon multiple (LCM).

LCM is always a natural number, which must be greater than the largest of the numbers for which it is defined.

Least common multiple (LCM). Properties.

Commutativity:

Associativity:

In particular, if and are coprime numbers , then:

Least common multiple of two integers m and n is a divisor of all other common multiples m and n. Moreover, the set of common multiples m,n coincides with the set of multiples for LCM( m,n).

The asymptotics for can be expressed in terms of some number-theoretic functions.

So, Chebyshev function. As well as:

This follows from the definition and properties of the Landau function g(n).

What follows from the law of distribution of prime numbers.

Finding the least common multiple (LCM).

NOC( a, b) can be calculated in several ways:

1. If the greatest common divisor is known, you can use its relationship with the LCM:

2. Let the canonical decomposition of both numbers into prime factors be known:

where p 1 ,...,p k are various prime numbers, and d 1 ,...,d k and e 1 ,...,ek are non-negative integers (they can be zero if the corresponding prime is not in the expansion).

Then LCM ( a,b) is calculated by the formula:

In other words, the LCM expansion contains all prime factors that are included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.

Example:

The calculation of the least common multiple of several numbers can be reduced to several successive calculations of the LCM of two numbers:

Rule. To find the LCM of a series of numbers, you need:

- decompose numbers into prime factors;

- transfer the largest expansion to the factors of the desired product (the product of the factors of the largest number of the given ones), and then add factors from the expansion of other numbers that do not occur in the first number or are in it a smaller number of times;

- the resulting product of prime factors will be the LCM of the given numbers.

Any two or more natural numbers have their own LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.

The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (the number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.

The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that all given numbers are multiples of.

The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.

rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.

Another option:

To find the least common multiple (LCM) of several numbers you need:

1) represent each number as a product of its prime factors, for example:

504 \u003d 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 \u003d 2 2 2 3 3 7 \u003d 2 3 3 2 7 1,

3) write down all prime divisors (multipliers) of each of these numbers;

4) choose the largest degree of each of them, found in all expansions of these numbers;

5) multiply these powers.

Example. Find the LCM of numbers: 168, 180 and 3024.

Solution. 168 \u003d 2 2 2 3 7 \u003d 2 3 3 1 7 1,

180 \u003d 2 2 3 3 5 \u003d 2 2 3 2 5 1,

3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1 .

We write out the largest powers of all prime divisors and multiply them:

LCM = 2 4 3 3 5 1 7 1 = 15120.

But many natural numbers are evenly divisible by other natural numbers.

For example:

The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;

The number 36 is divisible by 1, by 2, by 3, by 4, by 6, by 12, by 18, by 36.

The numbers by which the number is divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called number divisors. Divisor of a natural number a is the natural number that divides the given number a without a trace. A natural number that has more than two factors is called composite .

Note that the numbers 12 and 36 have common divisors. These are the numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. The common divisor of these two numbers a and b is the number by which both given numbers are divisible without a remainder a and b.

common multiple several numbers is called the number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all jcommon multiples, there is always the smallest one, in this case it is 90. This number is called leastcommon multiple (LCM).

LCM is always a natural number, which must be greater than the largest of the numbers for which it is defined.

Least common multiple (LCM). Properties.

Commutativity:

Associativity:

In particular, if and are coprime numbers , then:

Least common multiple of two integers m and n is a divisor of all other common multiples m and n. Moreover, the set of common multiples m,n coincides with the set of multiples for LCM( m,n).

The asymptotics for can be expressed in terms of some number-theoretic functions.

So, Chebyshev function. As well as:

This follows from the definition and properties of the Landau function g(n).

What follows from the law of distribution of prime numbers.

Finding the least common multiple (LCM).

NOC( a, b) can be calculated in several ways:

1. If the greatest common divisor is known, you can use its relationship with the LCM:

2. Let the canonical decomposition of both numbers into prime factors be known:

where p 1 ,...,p k are various prime numbers, and d 1 ,...,d k and e 1 ,...,ek are non-negative integers (they can be zero if the corresponding prime is not in the expansion).

Then LCM ( a,b) is calculated by the formula:

In other words, the LCM expansion contains all prime factors that are included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.

Example:

The calculation of the least common multiple of several numbers can be reduced to several successive calculations of the LCM of two numbers:

Rule. To find the LCM of a series of numbers, you need:

- decompose numbers into prime factors;

- transfer the largest expansion to the factors of the desired product (the product of the factors of the largest number of the given ones), and then add factors from the expansion of other numbers that do not occur in the first number or are in it a smaller number of times;

- the resulting product of prime factors will be the LCM of the given numbers.

Any two or more natural numbers have their own LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.

The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (the number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.

The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that all given numbers are multiples of.

The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.

rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.

Another option:

To find the least common multiple (LCM) of several numbers you need:

1) represent each number as a product of its prime factors, for example:

504 \u003d 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 \u003d 2 2 2 3 3 7 \u003d 2 3 3 2 7 1,

3) write down all prime divisors (multipliers) of each of these numbers;

4) choose the largest degree of each of them, found in all expansions of these numbers;

5) multiply these powers.

Example. Find the LCM of numbers: 168, 180 and 3024.

Solution. 168 \u003d 2 2 2 3 7 \u003d 2 3 3 1 7 1,

180 \u003d 2 2 3 3 5 \u003d 2 2 3 2 5 1,

3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1 .

We write out the largest powers of all prime divisors and multiply them:

LCM = 2 4 3 3 5 1 7 1 = 15120.

Finding the least common multiple (LCM) and the greatest common divisor (GCD) of natural numbers.

2

5

2

5

3

3

5

60=2*2*3*5
75=3*5*5
2) We write out the factors included in the expansion of the first of these numbers and add to them the missing factor 5 from the expansion of the second number. We get: 2*2*3*5*5=300. Found NOC, i.e. this sum = 300. Do not forget the dimension and write the answer:
Answer: Mom gives 300 rubles each.

Definition of GCD: Greatest Common Divisor (GCD) natural numbers a and in name the largest natural number c, to which and a, and b divided without remainder. Those. c is the smallest natural number for which and a and b are multiples.

Reminder: There are two approaches to the definition of natural numbers

  • numbers used in: enumeration (numbering) of items (first, second, third, ...); - in schools, usually.
  • indicating the number of items (no pokemon - zero, one pokemon, two pokemon, ...).

Negative and non-integer (rational, real, ...) numbers are not natural. Some authors include zero in the set of natural numbers, others do not. The set of all natural numbers is usually denoted by the symbol N

Reminder: Divisor of a natural number a call the number b, to which a divided without remainder. Multiple of natural number b called a natural number a, which is divided by b without a trace. If number b- number divisor a, then a multiple of b. Example: 2 is a divisor of 4 and 4 is a multiple of 2. 3 is a divisor of 12, and 12 is a multiple of 3.
Reminder: Natural numbers are called prime if they are divisible without remainder only by themselves and by 1. Coprime are numbers that have only one common divisor equal to 1.

Definition of how to find the GCD in the general case: To find GCD (Greatest Common Divisor) Several natural numbers are needed:
1) Decompose them into prime factors. (The Prime Number Chart can be very helpful for this.)
2) Write out the factors included in the expansion of one of them.
3) Delete those that are not included in the expansion of the remaining numbers.
4) Multiply the factors obtained in paragraph 3).

Task 2 on (NOK): By the new year, Kolya Puzatov bought 48 hamsters and 36 coffee pots in the city. Fekla Dormidontova, as the most honest girl in the class, was given the task of dividing this property into the largest possible number of gift sets for teachers. What is the number of sets? What is the composition of the sets?

Example 2.1. solving the problem of finding GCD. Finding GCD by selection.
Solution: Each of the numbers 48 and 36 must be divisible by the number of gifts.
1) Write out the divisors 48: 48, 24, 16, 12 , 8, 6, 3, 2, 1
2) Write out the divisors 36: 36, 18, 12 , 9, 6, 3, 2, 1 Choose the greatest common divisor. Op-la-la! Found, this is the number of sets of 12 pieces.
3) Divide 48 by 12, we get 4, divide 36 by 12, we get 3. Do not forget the dimension and write the answer:
Answer: You will get 12 sets of 4 hamsters and 3 coffee pots in each set.

A multiple of a number is a number that is divisible by a given number without a remainder. The least common multiple (LCM) of a group of numbers is the smallest number that is evenly divisible by each number in the group. To find the least common multiple, you need to find the prime factors of the given numbers. Also, LCM can be calculated using a number of other methods that are applicable to groups of two or more numbers.

Steps

A number of multiples

    Look at these numbers. The method described here is best used when given two numbers that are both less than 10. If large numbers are given, use a different method.

    • For example, find the least common multiple of the numbers 5 and 8. These are small numbers, so this method can be used.

  1. A multiple of a number is a number that is divisible by a given number without a remainder. Multiple numbers can be found in the multiplication table.

    • For example, numbers that are multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40.

  2. Write down a series of numbers that are multiples of the first number. Do this under multiples of the first number to compare two rows of numbers.

    • For example, numbers that are multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, and 64.

  3. Find the smallest number that appears in both series of multiples. You may have to write long series of multiples to find the total. The smallest number that appears in both series of multiples is the least common multiple.

    • For example, the smallest number that appears in the series of multiples of 5 and 8 is 40. Therefore, 40 is the least common multiple of 5 and 8.

    Prime factorization

    1. Look at these numbers. The method described here is best used when given two numbers that are both greater than 10. If smaller numbers are given, use a different method.

      • For example, find the least common multiple of the numbers 20 and 84. Each of the numbers is greater than 10, so this method can be used.

    2. Factorize the first number. That is, you need to find such prime numbers, when multiplied, you get a given number. Having found prime factors, write them down as an equality.

      • For example, 2 × 10 = 20 (\displaystyle (\mathbf (2) )\times 10=20) and 2 × 5 = 10 (\displaystyle (\mathbf (2) )\times (\mathbf (5) )=10). Thus, the prime factors of the number 20 are the numbers 2, 2 and 5. Write them down as an expression: .

    3. Factor the second number into prime factors. Do this in the same way as you factored the first number, that is, find such prime numbers that, when multiplied, will get this number.

      • For example, 2 × 42 = 84 (\displaystyle (\mathbf (2) )\times 42=84), 7 × 6 = 42 (\displaystyle (\mathbf (7) )\times 6=42) and 3 × 2 = 6 (\displaystyle (\mathbf (3) )\times (\mathbf (2) )=6). Thus, the prime factors of the number 84 are the numbers 2, 7, 3 and 2. Write them down as an expression: .

    4. Write down the factors common to both numbers. Write such factors as a multiplication operation. As you write down each factor, cross it out in both expressions (expressions that describe the decomposition of numbers into prime factors).

      • For example, the common factor for both numbers is 2, so write 2 × (\displaystyle 2\times ) and cross out the 2 in both expressions.
      • The common factor for both numbers is another factor of 2, so write 2 × 2 (\displaystyle 2\times 2) and cross out the second 2 in both expressions.

    5. Add the remaining factors to the multiplication operation. These are factors that are not crossed out in both expressions, that is, factors that are not common to both numbers.

      • For example, in the expression 20 = 2 × 2 × 5 (\displaystyle 20=2\times 2\times 5) both twos (2) are crossed out because they are common factors. The factor 5 is not crossed out, so write the multiplication operation as follows: 2 × 2 × 5 (\displaystyle 2\times 2\times 5)
      • In the expression 84 = 2 × 7 × 3 × 2 (\displaystyle 84=2\times 7\times 3\times 2) both deuces (2) are also crossed out. Factors 7 and 3 are not crossed out, so write the multiplication operation as follows: 2 × 2 × 5 × 7 × 3 (\displaystyle 2\times 2\times 5\times 7\times 3).

    6. Calculate the least common multiple. To do this, multiply the numbers in the written multiplication operation.

      • For example, 2 × 2 × 5 × 7 × 3 = 420 (\displaystyle 2\times 2\times 5\times 7\times 3=420). So the least common multiple of 20 and 84 is 420.

    Finding common divisors

    1. Draw a grid like you would for a game of tic-tac-toe. Such a grid consists of two parallel lines that intersect (at right angles) with two other parallel lines. This will result in three rows and three columns (the grid looks a lot like the # sign). Write the first number in the first row and second column. Write the second number in the first row and third column.

      • For example, find the least common multiple of 18 and 30. Write 18 in the first row and second column, and write 30 in the first row and third column.

    2. Find the divisor common to both numbers. Write it down in the first row and first column. It is better to look for prime divisors, but this is not a prerequisite.

      • For example, 18 and 30 are even numbers, so their common divisor is 2. So write 2 in the first row and first column.

    3. Divide each number by the first divisor. Write each quotient under the corresponding number. The quotient is the result of dividing two numbers.

      • For example, 18 ÷ 2 = 9 (\displaystyle 18\div 2=9), so write 9 under 18.
      • 30 ÷ 2 = 15 (\displaystyle 30\div 2=15), so write 15 under 30.

    4. Find a divisor common to both quotients. If there is no such divisor, skip the next two steps. Otherwise, write down the divisor in the second row and first column.

      • For example, 9 and 15 are divisible by 3, so write 3 in the second row and first column.

    5. Divide each quotient by the second divisor. Write each division result under the corresponding quotient.

      • For example, 9 ÷ 3 = 3 (\displaystyle 9\div 3=3), so write 3 under 9.
      • 15 ÷ 3 = 5 (\displaystyle 15\div 3=5), so write 5 under 15.

    6. If necessary, supplement the grid with additional cells. Repeat the above steps until the quotients have a common divisor.

    7. Circle the numbers in the first column and last row of the grid. Then write the highlighted numbers as a multiplication operation.

      • For example, the numbers 2 and 3 are in the first column, and the numbers 3 and 5 are in the last row, so write the multiplication operation like this: 2 × 3 × 3 × 5 (\displaystyle 2\times 3\times 3\times 5).

    8. Find the result of multiplying numbers. This will calculate the least common multiple of the two given numbers.

      • For example, 2 × 3 × 3 × 5 = 90 (\displaystyle 2\times 3\times 3\times 5=90). So the least common multiple of 18 and 30 is 90.

    Euclid's algorithm

    1. Remember the terminology associated with the division operation. The dividend is the number that is being divided. The divisor is the number by which to divide. The quotient is the result of dividing two numbers. The remainder is the number left when two numbers are divided.

      • For example, in the expression 15 ÷ 6 = 2 (\displaystyle 15\div 6=2) rest. 3:
        15 is the divisible
        6 is the divisor
        2 is private
        3 is the remainder.

To understand how to calculate the LCM, you should first determine the meaning of the term "multiple".


A multiple of A is a natural number that is divisible by A without remainder. Thus, 15, 20, 25, and so on can be considered multiples of 5.


There can be a limited number of divisors of a particular number, but there are an infinite number of multiples.


A common multiple of natural numbers is a number that is divisible by them without a remainder.

How to find the least common multiple of numbers

The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is evenly divisible by all these numbers.


To find the NOC, you can use several methods.


For small numbers, it is convenient to write out in a line all the multiples of these numbers until a common one is found among them. Multiples are denoted in the record with a capital letter K.


For example, multiples of 4 can be written like this:


K(4) = (8,12, 16, 20, 24, ...)


K(6) = (12, 18, 24, ...)


So, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This entry is performed as follows:


LCM(4, 6) = 24


If the numbers are large, find the common multiple of three or more numbers, then it is better to use another way to calculate the LCM.


To complete the task, it is necessary to decompose the proposed numbers into prime factors.


First you need to write out the expansion of the largest of the numbers in a line, and below it - the rest.


In the expansion of each number, there may be a different number of factors.


For example, let's factor the numbers 50 and 20 into prime factors.




In the expansion of the smaller number, one should underline the factors that are missing in the expansion of the first largest number, and then add them to it. In the presented example, a deuce is missing.


Now we can calculate the least common multiple of 20 and 50.


LCM (20, 50) = 2 * 5 * 5 * 2 = 100


Thus, the product of the prime factors of the larger number and the factors of the second number, which are not included in the decomposition of the larger number, will be the least common multiple.


To find the LCM of three or more numbers, all of them should be decomposed into prime factors, as in the previous case.


As an example, you can find the least common multiple of the numbers 16, 24, 36.


36 = 2 * 2 * 3 * 3


24 = 2 * 2 * 2 * 3


16 = 2 * 2 * 2 * 2


Thus, only two deuces from the decomposition of sixteen were not included in the factorization of a larger number (one is in the decomposition of twenty-four).


Thus, they need to be added to the decomposition of a larger number.


LCM (12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9


There are special cases of determining the least common multiple. So, if one of the numbers can be divided without a remainder by another, then the larger of these numbers will be the least common multiple.


For example, NOCs of twelve and twenty-four would be twenty-four.


If it is necessary to find the least common multiple of coprime numbers that do not have the same divisors, then their LCM will be equal to their product.


For example, LCM(10, 11) = 110.

RELATED ARTICLES