Least common multiple of 14 11. Divisors and multiples

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Find GCD and LOC

Found GCD and LOC: 5806

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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.

How to find LCM (least common multiple)

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.

Greatest common divisor

Definition 2

If a natural number a is divisible by a natural number $b$, then $b$ is called a divisor of $a$, and $a$ is called a multiple of $b$.

Let $a$ and $b$ be natural numbers. The number $c$ is called the common divisor of both $a$ and $b$.

The set of common divisors of the numbers $a$ and $b$ is finite, since none of these divisors can be greater than $a$. This means that among these divisors there is a largest one, which is called the greatest common divisor of the numbers $a$ and $b$ and is denoted by the following notation:

$GCD\(a;b)\ or \D\(a;b)$

To find the greatest common divisor of two numbers you need:

1. Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

Example 1

Find the gcd of the numbers $121$ and $132.$

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Choose the numbers that are included in the expansion of these numbers

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$GCD=2\cdot 11=22$

Example 2

Find the gcd of the monomials $63$ and $81$.

We will find according to the presented algorithm. For this:

Let's factor the numbers into prime factors

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

We select the numbers that are included in the expansion of these numbers

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

Let's find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$GCD=3\cdot 3=9$

You can find the gcd of two numbers in another way, using a set of number divisors.

Example 3

Find the gcd of the numbers $48$ and $60$.

Solution:

Let's find the set of divisors of the number $48$: $\left\((\rm 1,2,3.4.6,8,12,16,24,48)\right\)$

Now let's find the set of divisors of the number $60$:$\ \left\((\rm 1,2,3,4,5,6,10,12,15,20,30,60)\right\)$

Let's find the intersection of these sets: $\left\((\rm 1,2,3,4,6,12)\right\)$ - this set will determine the set of common divisors of the numbers $48$ and $60$. The largest element in this set will be the number $12$. This means that the greatest common divisor of the numbers $48$ and $60$ is $12$.

Definition of NPL

Definition 3

Common multiples of natural numbers$a$ and $b$ is a natural number that is a multiple of both $a$ and $b$.

Common multiples of numbers are numbers that are divisible by the original numbers without a remainder. For example, for the numbers $25$ and $50$, the common multiples will be the numbers $50,100,150,200$, etc.

The smallest common multiple will be called the least common multiple and will be denoted LCM$(a;b)$ or K$(a;b).$

To find the LCM of two numbers, you need to:

1. Factor numbers into prime factors
2. Write down the factors that are part of the first number and add to them the factors that are part of the second and are not part of the first

Example 4

Find the LCM of the numbers $99$ and $77$.

We will find according to the presented algorithm. For this

Factor numbers into prime factors

$99=3\cdot 3\cdot 11$

Write down the factors included in the first

add to them multipliers that are part of the second and not part of the first

Find the product of the numbers found in step 2. The resulting number will be the desired least common multiple

$NOK=3\cdot 3\cdot 11\cdot 7=693$

Compiling lists of divisors of numbers is often a very labor-intensive task. There is a way to find GCD called the Euclidean algorithm.

Statements on which the Euclidean algorithm is based:

If $a$ and $b$ are natural numbers, and $a\vdots b$, then $D(a;b)=b$

If $a$ and $b$ are natural numbers such that $b

Using $D(a;b)= D(a-b;b)$, we can successively reduce the numbers under consideration until we reach a pair of numbers such that one of them is divisible by the other. Then the smaller of these numbers will be the desired greatest common divisor for the numbers $a$ and $b$.

Properties of GCD and LCM

1. Any common multiple of $a$ and $b$ is divisible by K$(a;b)$
2. If $a\vdots b$ , then К$(a;b)=a$

If K$(a;b)=k$ and $m$ is a natural number, then K$(am;bm)=km$

If $d$ is a common divisor for $a$ and $b$, then K($\frac(a)(d);\frac(b)(d)$)=$\ \frac(k)(d) $

If $a\vdots c$ and $b\vdots c$ , then $\frac(ab)(c)$ is the common multiple of $a$ and $b$

For any natural numbers $a$ and $b$ the equality holds

$D(a;b)\cdot К(a;b)=ab$

Any common divisor of the numbers $a$ and $b$ is a divisor of the number $D(a;b)$

But many natural numbers are also 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 by a whole (for 12 these are 1, 2, 3, 4, 6 and 12) are called divisors of numbers. Divisor of a natural number a- is a natural number that divides a given number a without a trace. A natural number that has more than two divisors is called composite .

Please note that the numbers 12 and 36 have common factors. These numbers are: 1, 2, 3, 4, 6, 12. The greatest divisor of these numbers is 12. The common divisor of these two numbers a And b- this is the number by which both given numbers are divided without remainder a And b.

Common multiples several numbers is a 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 common multiples there is always a smallest one, in this case it is 90. This number is called the smallestcommon multiple (CMM).

The LCM is always a natural number that 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 of the LCM( m, n).

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

So, Chebyshev function. And:

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 connection with the LCM:

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

Where p 1 ,...,p k- various prime numbers, and d 1 ,...,d k And e 1 ,...,e k— non-negative integers (they can be zeros if the corresponding prime is not in the expansion).

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

In other words, the LCM decomposition contains all prime factors included in at least one of the decompositions of numbers a, b, and the largest of the two exponents of this multiplier is taken.

Example:

Calculating the least common multiple of several numbers can be reduced to several sequential 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 decomposition (the product of the factors of the largest number of the given ones) to the factors of the desired product, and then add factors from the decomposition of other numbers that do not appear in the first number or appear in it fewer 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) are 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 are supplemented by the factor 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 is a multiple of all given numbers.

The numbers 2,3,11,37 are prime numbers, 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 = 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,

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

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

5) multiply these powers.

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

Solution. 168 = 2 2 2 3 7 = 2 3 3 1 7 1,

180 = 2 2 3 3 5 = 2 2 3 2 5 1,

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

We write down the greatest powers of all prime divisors and multiply them:

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

A multiple 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 divisible by each number in the group without leaving a remainder. To find the least common multiple, you need to find the prime factors of given numbers. The LCM can also be calculated using a number of other methods that apply to groups of two or more numbers.

Steps

Series of multiples

Look at these numbers. The method described here is best used when given two numbers, each of which is less than 10. If larger numbers are given, use a different method.

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

1. A multiple is a number that is divisible by a given number without a remainder. Multiples 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 sets 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 is present in both sets of multiples. You may have to write long series of multiples to find the total number. The smallest number that is present in both sets of multiples is the least common multiple.

   • For example, the smallest number that appears in the series of multiples of 5 and 8 is the number 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, each of which is 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 you can use this method.

   2. Factor the first number into prime factors. That is, you need to find such prime numbers that, when multiplied, will result in a given number. Once you have found the prime factors, write them as equalities.

      • 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 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 yield the given 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 as an expression: .

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

      • For example, both numbers have a common factor of 2, so write 2 × (\displaystyle 2\times ) and cross out the 2 in both expressions.
      • What both numbers have in common 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 like this: 2 × 2 × 5 (\displaystyle 2\times 2\times 5)
      • In expression 84 = 2 × 7 × 3 × 2 (\displaystyle 84=2\times 7\times 3\times 2) both twos (2) are also crossed out. The factors 7 and 3 are not crossed out, so write the multiplication operation like this: 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 factors

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

      • For example, find the least common multiple of the numbers 18 and 30. Write the number 18 in the first row and second column, and write the number 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 factors, but this is not a requirement.

      • For example, 18 and 30 are even numbers, so their common factor 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 appropriate number. A 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 down 15 under 30.

   4. Find the divisor common to both quotients. If there is no such divisor, skip the next two steps. Otherwise, write 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 its 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, add additional cells to the grid. Repeat the described 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 selected 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 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 that is being divided by. A quotient is the result of dividing two numbers. A remainder is the number left when two numbers are divided.

      • For example, in the expression 15 ÷ 6 = 2 (\displaystyle 15\div 6=2) ost. 3:
        15 is the dividend
        6 is a divisor
        2 is quotient
        3 is the remainder.