Many divisors

Consider the following problem: find the divisor of the number 140. It is obvious that the number 140 has not one divisor, but several. In such cases, the task is said to have a bunch of solutions. Let's find them all. First of all, we decompose this number into prime factors:

140 = 2 ∙ 2 ∙ 5 ∙ 7.

Now we can easily write out all the divisors. Let's start with simple divisors, that is, those that are present in the expansion above:

Then we write out those that are obtained by pairwise multiplication of prime divisors:

2∙2 = 4, 2∙5 = 10, 2∙7 = 14, 5∙7 = 35.

Then - those that contain three simple divisors:

2∙2∙5 = 20, 2∙2∙7 = 28, 2∙5∙7 = 70.

Finally, let's not forget the unit and the decomposable number itself:

All divisors found by us form a bunch of divisors of the number 140, which is written using curly braces:

The set of divisors of the number 140 =

{1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140}.

For convenience of perception, we have written out the divisors here ( set elements) in ascending order, but generally speaking, this is not necessary. In addition, we introduce an abbreviation. Instead of "The set of divisors of the number 140" we will write "D (140)". Thus,

Similarly, one can find the set of divisors for any other natural number. For example, from the expansion

105 = 3 ∙ 5 ∙ 7

we get:

D(105) = (1, 3, 5, 7, 15, 21, 35, 105).

From the set of all divisors, one should distinguish the set of prime divisors, which for the numbers 140 and 105 are equal, respectively:

PD(140) = (2, 5, 7).

PD(105) = (3, 5, 7).

It should be emphasized that in the decomposition of the number 140 into prime factors, two is present twice, while in the set PD(140) it is only one. The set of PD(140) is, in essence, all the answers to the problem: "Find a prime factor of the number 140". It is clear that the same answer should not be repeated more than once.

Fraction reduction. Greatest Common Divisor

Consider a fraction

We know that this fraction can be reduced by a number that is both a divisor of the numerator (105) and a divisor of the denominator (140). Let's look at the sets D(105) and D(140) and write down their common elements.

D(105) = (1, 3, 5, 7, 15, 21, 35, 105);

D(140) = (1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140).

Common elements of the sets D(105) and D(140) =

The last equality can be written shorter, namely:

D(105) ∩ D(140) = (1, 5, 7, 35).

Here, the special icon "∩" ("bag with the hole down") just indicates that from the two sets written on opposite sides of it, only common elements should be selected. The entry "D (105) ∩ D (140)" reads " intersection sets of Te from 105 and Te from 140.

[Note along the way that you can perform various binary operations with sets, almost like with numbers. Another common binary operation is Union, which is indicated by the icon "∪" ("bag with the hole up"). The union of two sets includes all the elements of both sets:

PD(105) = (3, 5, 7);

PD(140) = (2, 5, 7);

PD(105) ∪ PD(140) = (2, 3, 5, 7). ]

So, we found out that the fraction

can be reduced to any of the numbers belonging to the set

D(105) ∩ D(140) = (1, 5, 7, 35)

and cannot be reduced by any other natural number. Here are all possible ways to reduce (except for the uninteresting reduction by one):

It is obvious that it is most practical to reduce the fraction by a number, if possible, a larger one. In this case, it is the number 35, which is said to be greatest common divisor (GCD) numbers 105 and 140. This is written as

gcd(105, 140) = 35.

However, in practice, if we are given two numbers and need to find their greatest common divisor, we do not have to build any sets at all. It is enough to simply factor both numbers into prime factors and underline those of these factors that are common to both factorizations, for example:

105 = 3 ∙ 5 ∙ 7 ;

140 = 2 ∙ 2 ∙ 5 ∙ 7 .

Multiplying the underlined numbers (in any of the expansions), we get:

gcd(105, 140) = 5 ∙ 7 = 35.

Of course, it is possible that there are more than two underlined factors:

168 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 7;

396 = 2 ∙ 2 ∙ 3 ∙ 3 ∙ 11.

From here it is clear that

gcd(168, 396) = 2 ∙ 2 ∙ 3 = 12.

Special mention deserves the situation when there are no common factors at all and there is nothing to emphasize, for example:

42 = 2 ∙ 3 ∙ 7;

In this case,

gcd(42, 55) = 1.

Two natural numbers for which the gcd is equal to one are called coprime. If you make a fraction from such numbers, for example,

then such a fraction is irreducible.

Generally speaking, the rule for reducing fractions can be written as follows:

		a/ gcd( a, b)
		b/ gcd( a, b)

Here it is assumed that a and b are natural numbers, and all fractions are positive. If we now assign a minus sign to both sides of this equality, we get the corresponding rule for negative fractions.

Addition and subtraction of fractions. Least common multiple

Suppose you want to calculate the sum of two fractions:

We already know how denominators are decomposed into prime factors:

105 = 3 ∙ 5 ∙ 7 ;

140 = 2 ∙ 2 ∙ 5 ∙ 7 .

It immediately follows from this decomposition that, in order to bring the fractions to a common denominator, it is enough to multiply the numerator and denominator of the first fraction by 2 ∙ 2 (the product of unstressed prime factors of the second denominator), and the numerator and denominator of the second fraction by 3 (“product” ununderlined prime factors of the first denominator). As a result, the denominators of both fractions will become equal to a number that can be represented as follows:

2 ∙ 2 ∙ 3 ∙ 5 ∙ 7 = 105 ∙ 2 ∙ 2 = 140 ∙ 3 = 420.

It is easy to see that both original denominators (both 105 and 140) are divisors of the number 420, and the number 420, in turn, is a multiple of both denominators - and not just a multiple, it is least common multiple (NOC) numbers 105 and 140. This is written like this:

LCM(105, 140) = 420.

Looking more closely at the expansion of the numbers 105 and 140, we see that

105 ∙ 140 = LCM(105, 140) ∙ GCD(105, 140).

Similarly, for arbitrary natural numbers b and d:

b ∙ d= LCM( b, d) ∙ GCD( b, d).

Now let's complete the summation of our fractions:

3 ∙ 5 ∙ 7		2 ∙ 2 ∙ 5 ∙ 7

2 ∙ 2 ∙ 3 ∙ 5 ∙ 7		2 ∙ 2 ∙ 3 ∙ 5 ∙ 7

2 ∙ 2 ∙ 3 ∙ 5 ∙ 7

2 ∙ 2 ∙ 3 ∙ 5 ∙ 7

2 ∙ 2 ∙ 3 ∙ 5

Note. To solve some problems, you need to know what the square of a number is. Number square a called a number a multiplied by itself, that is a∙a. (As you can see, it is equal to the area of ​​a square with a side a).

Signs of divisibility of natural numbers.

Numbers divisible by 2 without remainder are calledeven .

Numbers that are not evenly divisible by 2 are calledodd .

Sign of divisibility by 2

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

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

Sign of divisibility by 3

If the sum of the digits of a number is divisible by 3, then the number is also 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 if the number 2772825 is divisible by 3. To do this, we calculate the sum of the digits of this number: 2+7+7+2+8+2+5 = 33 - is divisible by 3. So, the number 2772825 is divisible by 3.

Sign of divisibility by 5

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

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

Sign of divisibility by 9

If the sum of the digits of a number is divisible by 9, then the number is also 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 if the number 5402070 is divisible by 9. To do this, we calculate the sum of the digits of this number: 5+4+0+2+0+7+0 = 16 - is not divisible by 9. This means that the number 5402070 is not divisible by 9.

Sign of divisibility by 10

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

For example, the numbers 40 , 17 0 , 1409 0 are divisible without remainder by 10, and the numbers 17 , 9 3 , 1430 7 - do not 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. We decompose 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

There are factors 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) decompose them into prime factors;

2) write out 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 LCM (75;60). Let's use the rule.

1. We decompose the numbers 75 and 60 into prime factors.

75 = 3 · 5 · 5

60 = 2 · 2 · 3 · 3

2. Write down the factors included in the expansion of the number 75: 3, 5, 5.

NOC (75; 60) = 3 · 5 · 5 · …

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

NOC (75; 60) = 3 · 5 · 5 · 2 · 2

4. Find the product of the resulting factors

NOC (75; 60) = 3 · 5 · 5 · 2 · 2 = 300.

Let's solve the problem. We have two types of cookies. Some are chocolate and some are plain. There are 48 chocolate pieces, and simple 36. It is necessary to make the maximum possible number of gifts from these cookies, and all of them must be used.

First, let's write down all the divisors of each of these two numbers, since both of these numbers must be divisible by the number of gifts.

We get

• 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
• 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Let's find among the divisors the common ones that both the first and the second number have.

Common divisors will be: 1, 2, 3, 4, 6, 12.

The greatest common divisor of all is 12. This number is called the greatest common divisor of 36 and 48.

Based on the result, we can conclude that 12 gifts can be made from all cookies. One such gift will contain 4 chocolate cookies and 3 regular cookies.

Finding the Greatest Common Divisor

• The largest natural number by which two numbers a and b are divisible without a remainder is called the greatest common divisor of these numbers.

Sometimes the abbreviation GCD is used to abbreviate the entry.

Some pairs of numbers have one as their greatest common divisor. Such numbers are called coprime numbers. For example, numbers 24 and 35. Have GCD =1.

How to find the greatest common divisor

In order to find the greatest common divisor, it is not necessary to write out all the divisors of these numbers.

You can do otherwise. First, factor both numbers into prime factors.

• 48 = 2*2*2*2*3,
• 36 = 2*2*3*3.

Now, from the factors that are included in the expansion of the first number, we delete all those that are not included in the expansion of the second number. In our case, these are two deuces.

• 48 = 2*2*2*2*3 ,
• 36 = 2*2*3 *3.

The factors 2, 2 and 3 remain. Their product is 12. This number will be the greatest common divisor of the numbers 48 and 36.

This rule can be extended to the case of three, four, and so on. numbers.

General scheme for finding the greatest common divisor

• 1. Decompose numbers into prime factors.
• 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. Calculate the product of the remaining factors.

To learn how to find the greatest common divisor of two or more numbers, you need to understand what natural, prime and complex numbers are.

A natural number is any number that is used to count integers.

If a natural number can only be divided by itself and one, then it is called prime.

All natural numbers can be divided by themselves and one, but the only even prime number is 2, all other primes can be divided by two. Therefore, only odd numbers can be prime.

There are a lot of prime numbers, there is no complete list of them. To find the GCD, it is convenient to use special tables with such numbers.

Most natural numbers can be divided not only by one, themselves, but also by other numbers. So, for example, the number 15 can be divided by 3 and 5. All of them are called divisors of the number 15.

Thus, the divisor of any A is the number by which it can be divided without a remainder. If a number has more than two natural divisors, it is called composite.

The number 30 has such divisors as 1, 3, 5, 6, 15, 30.

You can see that 15 and 30 have the same divisors 1, 3, 5, 15. The greatest common divisor of these two numbers is 15.

Thus, the common divisor of the numbers A and B is the number by which you can divide them completely. The maximum can be considered the maximum total number by which they can be divided.

To solve problems, the following abbreviated inscription is used:

GCD (A; B).

For example, GCD (15; 30) = 30.

To write down all divisors of a natural number, the notation is used:

D(15) = (1, 3, 5, 15)

gcd (9; 15) = 1

In this example, natural numbers have only one common divisor. They are called coprime, respectively, the unit is their greatest common divisor.

How to find the greatest common divisor of numbers

To find the GCD of several numbers, you need:

Find all divisors of each natural number separately, that is, decompose them into factors (prime numbers);

Select all the same factors for given numbers;

Multiply them together.

For example, to calculate the greatest common divisor of the numbers 30 and 56, you would write the following:

In order not to get confused with , it is convenient to write the multipliers using vertical columns. On the left side of the line, you need to place the dividend, and on the right - the divisor. Under the dividend, you should indicate the resulting quotient.

So, in the right column will be all the factors needed for the solution.

Identical divisors (factors found) can be underlined for convenience. They should be rewritten and multiplied and the greatest common divisor should be written down.

GCD (30; 56) = 2 * 5 = 10

It's really that simple to find the greatest common divisor of numbers. With a little practice, you can do it almost automatically.

The largest natural number by which the numbers a and b are divisible without remainder is called greatest common divisor these numbers. Denote GCD(a, b).

Consider finding the GCD using the example of two natural numbers 18 and 60:

1 Let's decompose the numbers into prime factors:
18 = 2×3×3
60 = 2×2×3×5

2 Delete from the expansion of the first number all factors that are not included in the expansion of the second number, we get 2×3×3 .

3 We multiply the remaining prime factors after crossing out and get the greatest common divisor of numbers: gcd ( 18 , 60 )=2×3= 6 .

4 Note that it doesn’t matter from the first or second number we cross out the factors, the result will be the same:
18 = 2×3×3
60 = 2×2×3×5

324 , 111 and 432

Let's decompose the numbers into prime factors:

324 = 2×2×3×3×3×3

111 = 3×37

432 = 2×2×2×2×3×3×3

Delete from the first number, the factors of which are not in the second and third numbers, we get:

2 x 2 x 2 x 2 x 3 x 3 x 3 = 3

As a result of GCD( 324 , 111 , 432 )=3

Finding GCD with Euclid's Algorithm

The second way to find the greatest common divisor using Euclid's algorithm. Euclid's algorithm is the most efficient way to find GCD, using it you need to constantly find the remainder of the division of numbers and apply recurrent formula.

Recurrent formula for GCD, gcd(a, b)=gcd(b, a mod b), where a mod b is the remainder of dividing a by b.

Euclid's algorithm

Example Find the Greatest Common Divisor of Numbers 7920 and 594

Let's find GCD( 7920 , 594 ) using the Euclid algorithm, we will calculate the remainder of the division using a calculator.

GCD( 7920 , 594 )

GCD( 594 , 7920 mod 594 ) = gcd( 594 , 198 )

GCD( 198 , 594 mod 198 ) = gcd( 198 , 0 )

GCD( 198 , 0 ) = 198

• 7920 mod 594 = 7920 - 13 × 594 = 198
• 594 mod 198 = 594 - 3 × 198 = 0

As a result, we get GCD( 7920 , 594 ) = 198

Least common multiple

In order to find a common denominator when adding and subtracting fractions with different denominators, you need to know and be able to calculate least common multiple(NOC).

A multiple of the number "a" is a number that is itself divisible by the number "a" without a remainder.

Numbers that are multiples of 8 (that is, these numbers will be divided by 8 without a remainder): these are the numbers 16, 24, 32 ...

Multiples of 9: 18, 27, 36, 45…

There are infinitely many multiples of a given number a, in contrast to the divisors of the same number. Divisors - a finite number.

A common multiple of two natural numbers is a number that is evenly divisible by both of these numbers..

Least common multiple(LCM) of two or more natural numbers is the smallest natural number that is itself divisible by each of these numbers.

How to find the NOC

LCM can be found and written in two ways.

The first way to find the LCM

This method is usually used for small numbers.

1. We write the multiples for each of the numbers in a line until there is a multiple that is the same for both numbers.
2. A multiple of the number "a" is denoted by a capital letter "K".

Example. Find LCM 6 and 8.

The second way to find the LCM

This method is convenient to use to find the LCM for three or more numbers.

The number of identical factors in the expansions of numbers can be different.

In the expansion of the smaller number (smaller numbers), underline the factors that were not included in the expansion of the larger number (in our example, it is 2) and add these factors to the expansion of the larger number.
LCM (24, 60) = 2 2 3 5 2

Record the resulting work in response.
Answer: LCM (24, 60) = 120

You can also formalize finding the least common multiple (LCM) as follows. Let's find the LCM (12, 16, 24) .

24 = 2 2 2 3

As you can see from the expansion of numbers, all factors of 12 are included in the expansion of 24 (the largest of the numbers), so we add only one 2 from the expansion of the number 16 to the LCM.

LCM (12, 16, 24) = 2 2 2 3 2 = 48

Answer: LCM (12, 16, 24) = 48

Special cases of finding NOCs

If one of the numbers is evenly divisible by the others, then the least common multiple of these numbers is equal to this number.

For example, LCM(60, 15) = 60
Since coprime numbers have no common prime divisors, their least common multiple is equal to the product of these numbers.

On our site, you can also use a special calculator to find the least common multiple online to check your calculations.

If a natural number is only divisible by 1 and itself, then it is called prime.

Any natural number is always divisible by 1 and itself.

The number 2 is the smallest prime number. This is the only even prime number, the rest of the prime numbers are odd.

There are many prime numbers, and the first among them is the number 2. However, there is no last prime number. In the "For Study" section, you can download a table of prime numbers up to 997.

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

• the number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
• 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 evenly divisible (for 12 these are 1, 2, 3, 4, 6 and 12) are called the divisors of the number.

The divisor of a natural number a is such a natural number that divides the given number "a" without a remainder.

A natural number that has more than two factors is called a composite number.

Note that the numbers 12 and 36 have common divisors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12.

The common divisor of two given numbers "a" and "b" is the number by which both given numbers "a" and "b" are divided without remainder.

Greatest Common Divisor(GCD) of two given numbers "a" and "b" is the largest number by which both numbers "a" and "b" are divisible without a remainder.

Briefly, the greatest common divisor of numbers "a" and "b" is written as follows:

Example: gcd (12; 36) = 12 .

The divisors of numbers in the solution record are denoted by a capital letter "D".

The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.

Coprime numbers are natural numbers that have only one common divisor - the number 1. Their GCD is 1.

How to find the greatest common divisor

To find the gcd of two or more natural numbers you need:

• decompose the divisors of numbers into prime factors;

Calculations are conveniently written using a vertical bar. To the left of the line, first write down the dividend, to the right - the divisor. Further in the left column we write down the values ​​of private.

Let's explain right away with an example. Let's factorize the numbers 28 and 64 into prime factors.

Underline the same prime factors in both numbers.
28 = 2 2 7

64 = 2 2 2 2 2 2
We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4

Answer: GCD (28; 64) = 4

You can arrange the location of the GCD in two ways: in a column (as was done above) or “in a line”.

The first way to write GCD

Find GCD 48 and 36.

GCD (48; 36) = 2 2 3 = 12

The second way to write GCD

Now let's write the GCD search solution in a line. Find GCD 10 and 15.

On our information site, you can also find the greatest common divisor online using the helper program to check your calculations.

Finding the least common multiple, methods, examples of finding the LCM.

The material presented below is a logical continuation of the theory from the article under the heading LCM - Least Common Multiple, definition, examples, relationship between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and pay special attention to solving examples. Let us first show how the LCM of two numbers is calculated in terms of the GCD of these numbers. Next, consider finding the least common multiple by factoring numbers into prime factors. After that, we will focus on finding the LCM of three or more numbers, and also pay attention to the calculation of the LCM of negative numbers.

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Calculation of the least common multiple (LCM) through gcd

One way to find the least common multiple is based on the relationship between LCM and GCD. The existing relationship between LCM and GCD allows you to calculate the least common multiple of two positive integers through the known greatest common divisor. The corresponding formula has the form LCM(a, b)=a b: GCM(a, b). Consider examples of finding the LCM according to the above formula.

Find the least common multiple of the two numbers 126 and 70 .

In this example a=126 , b=70 . Let's use the link of LCM with GCD, which is expressed by the formula LCM(a, b)=a b: GCM(a, b) . That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers according to the written formula.

Find gcd(126, 70) using Euclid's algorithm: 126=70 1+56 , 70=56 1+14 , 56=14 4 , hence gcd(126, 70)=14 .

Now we find the required least common multiple: LCM(126, 70)=126 70:GCD(126, 70)= 126 70:14=630 .

What is LCM(68, 34) ?

Since 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34:GCD(68, 34)= 68 34:34=68 .

Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b , then the least common multiple of these numbers is a .

Finding the LCM by Factoring Numbers into Prime Factors

Another way to find the least common multiple is based on factoring numbers into prime factors. If we make a product of all prime factors of these numbers, after which we exclude from this product all common prime factors that are present in the expansions of these numbers, then the resulting product will be equal to the least common multiple of these numbers.

The announced rule for finding the LCM follows from the equality LCM(a, b)=a b: GCD(a, b) . Indeed, the product of the numbers a and b is equal to the product of all the factors involved in the expansions of the numbers a and b. In turn, gcd(a, b) is equal to the product of all prime factors that are simultaneously present in the expansions of the numbers a and b (which is described in the section on finding the gcd using the decomposition of numbers into prime factors).

Let's take an example. Let we know that 75=3 5 5 and 210=2 3 5 7 . Compose the product of all factors of these expansions: 2 3 3 5 5 5 7 . Now we exclude from this product all the factors that are present both in the expansion of the number 75 and in the expansion of the number 210 (such factors are 3 and 5), then the product will take the form 2 3 5 5 7 . The value of this product is equal to the least common multiple of 75 and 210 , that is, LCM(75, 210)= 2 3 5 5 7=1 050 .

After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.

Let's decompose the numbers 441 and 700 into prime factors:

We get 441=3 3 7 7 and 700=2 2 5 5 7 .

Now let's make a product of all the factors involved in the expansions of these numbers: 2 2 3 3 5 5 7 7 7 . Let us exclude from this product all the factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2 2 3 3 5 5 7 7 . So LCM(441, 700)=2 2 3 3 5 5 7 7=44 100 .

LCM(441, 700)= 44 100 .

The rule for finding the LCM using the decomposition of numbers into prime factors can be formulated a little differently. If we add the missing factors from the expansion of the number b to the factors from the expansion of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.

For example, let's take all the same numbers 75 and 210, their expansions into prime factors are as follows: 75=3 5 5 and 210=2 3 5 7 . To the factors 3, 5 and 5 from the decomposition of the number 75, we add the missing factors 2 and 7 from the decomposition of the number 210, we get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .

Find the least common multiple of 84 and 648.

We first obtain the decomposition of the numbers 84 and 648 into prime factors. They look like 84=2 2 3 7 and 648=2 2 2 3 3 3 3 . To the factors 2 , 2 , 3 and 7 from the decomposition of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the decomposition of the number 648 , we get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 84 and 648 is 4,536.

Finding the LCM of three or more numbers

The least common multiple of three or more numbers can be found by successively finding the LCM of two numbers. Recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.

Let positive integers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k =LCM(m k−1 , a k) .

Consider the application of this theorem on the example of finding the least common multiple of four numbers.

Find the LCM of the four numbers 140 , 9 , 54 and 250 .

First we find m 2 = LCM (a 1 , a 2) = LCM (140, 9) . To do this, using the Euclidean algorithm, we determine gcd(140, 9) , we have 140=9 15+5 , 9=5 1+4 , 5=4 1+1 , 4=1 4 , therefore, gcd(140, 9)=1 , whence LCM(140, 9)=140 9: GCD(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .

Now we find m 3 = LCM (m 2 , a 3) = LCM (1 260, 54) . Let's calculate it through gcd(1 260, 54) , which is also determined by the Euclid algorithm: 1 260=54 23+18 , 54=18 3 . Then gcd(1 260, 54)=18 , whence LCM(1 260, 54)= 1 260 54:gcd(1 260, 54)= 1 260 54:18=3 780 . That is, m 3 \u003d 3 780.

It remains to find m 4 = LCM (m 3 , a 4) = LCM (3 780, 250) . To do this, we find GCD(3 780, 250) using the Euclid algorithm: 3 780=250 15+30 , 250=30 8+10 , 30=10 3 . Therefore, gcd(3 780, 250)=10 , hence LCM(3 780, 250)= 3 780 250:gcd(3 780, 250)= 3 780 250:10=94 500 . That is, m 4 \u003d 94 500.

So the least common multiple of the original four numbers is 94,500.

LCM(140, 9, 54, 250)=94500 .

In many cases, the least common multiple of three or more numbers is conveniently found using prime factorizations of given numbers. In this case, the following rule should be followed. The least common multiple of several numbers is equal to the product, which is composed as follows: the missing factors from the expansion of the second number are added to all the factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the obtained factors, and so on.

Consider an example of finding the least common multiple using the decomposition of numbers into prime factors.

Find the least common multiple of five numbers 84 , 6 , 48 , 7 , 143 .

First, we obtain decompositions of these numbers into prime factors: 84=2 2 3 7 , 6=2 3 , 48=2 2 2 2 3 , 7 (7 is a prime number, it coincides with its decomposition into prime factors) and 143=11 13 .

To find the LCM of these numbers, to the factors of the first number 84 (they are 2 , 2 , 3 and 7) you need to add the missing factors from the expansion of the second number 6 . The expansion of the number 6 does not contain missing factors, since both 2 and 3 are already present in the expansion of the first number 84 . Further to the factors 2 , 2 , 3 and 7 we add the missing factors 2 and 2 from the expansion of the third number 48 , we get a set of factors 2 , 2 , 2 , 2 , 3 and 7 . There is no need to add factors to this set in the next step, since 7 is already contained in it. Finally, to the factors 2 , 2 , 2 , 2 , 3 and 7 we add the missing factors 11 and 13 from the expansion of the number 143 . We get the product 2 2 2 2 3 7 11 13 , which is equal to 48 048 .

Therefore, LCM(84, 6, 48, 7, 143)=48048 .

LCM(84, 6, 48, 7, 143)=48048 .

Finding the Least Common Multiple of Negative Numbers

Sometimes there are tasks in which you need to find the least common multiple of numbers, among which one, several or all numbers are negative. In these cases, all negative numbers must be replaced by their opposite numbers, after which the LCM of positive numbers should be found. This is the way to find the LCM of negative numbers. For example, LCM(54, −34)=LCM(54, 34) and LCM(−622, −46, −54, −888)= LCM(622, 46, 54, 888) .

We can do this because the set of multiples of a is the same as the set of multiples of −a (a and −a are opposite numbers). Indeed, let b be some multiple of a , then b is divisible by a , and the concept of divisibility asserts the existence of such an integer q that b=a q . But the equality b=(−a)·(−q) will also be true, which, by virtue of the same concept of divisibility, means that b is divisible by −a , that is, b is a multiple of −a . The converse statement is also true: if b is some multiple of −a , then b is also a multiple of a .

Find the least common multiple of the negative numbers −145 and −45.

Let's replace the negative numbers −145 and −45 with their opposite numbers 145 and 45 . We have LCM(−145, −45)=LCM(145, 45) . Having determined gcd(145, 45)=5 (for example, using the Euclid algorithm), we calculate LCM(145, 45)=145 45:gcd(145, 45)= 145 45:5=1 305 . Thus, the least common multiple of the negative integers −145 and −45 is 1,305 .

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We continue to study division. In this lesson, we will look at concepts such as GCD and NOC.

GCD is the greatest common divisor.

NOC is the least common multiple.

The topic is rather boring, but it is necessary to understand it. Without understanding this topic, you will not be able to work effectively with fractions, which are a real obstacle in mathematics.

Greatest Common Divisor

Definition. Greatest Common Divisor of Numbers a and b a and b divided without remainder.

In order to understand this definition well, we substitute instead of variables a and b any two numbers, for example, instead of a variable a substitute the number 12, and instead of the variable b number 9. Now let's try to read this definition:

Greatest Common Divisor of Numbers 12 and 9 is the largest number by which 12 and 9 divided without remainder.

It is clear from the definition that we are talking about a common divisor of the numbers 12 and 9, and this divisor is the largest of all existing divisors. This greatest common divisor (gcd) must be found.

To find the greatest common divisor of two numbers, three methods are used. The first method is quite time-consuming, but it allows you to understand the essence of the topic well and feel its whole meaning.

The second and third methods are quite simple and make it possible to quickly find the GCD. We will consider all three methods. And what to apply in practice - you choose.

The first way is to find all possible divisors of two numbers and choose the largest of them. Let's consider this method in the following example: find the greatest common divisor of the numbers 12 and 9.

First, we find all possible divisors of the number 12. To do this, we divide 12 into all divisors in the range from 1 to 12. If the divisor allows us to divide 12 without a remainder, then we will highlight it in blue and make an appropriate explanation in brackets.

12: 1 = 12
(12 divided by 1 without a remainder, so 1 is a divisor of 12)

12: 2 = 6
(12 divided by 2 without a remainder, so 2 is a divisor of 12)

12: 3 = 4
(12 divided by 3 without a remainder, so 3 is a divisor of 12)

12: 4 = 3
(12 divided by 4 without a remainder, so 4 is a divisor of 12)

12:5 = 2 (2 left)
(12 is not divided by 5 without a remainder, so 5 is not a divisor of 12)

12: 6 = 2
(12 divided by 6 without a remainder, so 6 is a divisor of 12)

12: 7 = 1 (5 left)
(12 is not divided by 7 without a remainder, so 7 is not a divisor of 12)

12: 8 = 1 (4 left)
(12 is not divided by 8 without a remainder, so 8 is not a divisor of 12)

12:9 = 1 (3 left)
(12 is not divided by 9 without a remainder, so 9 is not a divisor of 12)

12: 10 = 1 (2 left)
(12 is not divided by 10 without a remainder, so 10 is not a divisor of 12)

12:11 = 1 (1 left)
(12 is not divided by 11 without a remainder, so 11 is not a divisor of 12)

12: 12 = 1
(12 divided by 12 without a remainder, so 12 is a divisor of 12)

Now let's find the divisors of the number 9. To do this, check all the divisors from 1 to 9

9: 1 = 9
(9 divided by 1 without a remainder, so 1 is a divisor of 9)

9: 2 = 4 (1 left)
(9 is not divided by 2 without a remainder, so 2 is not a divisor of 9)

9: 3 = 3
(9 divided by 3 without a remainder, so 3 is a divisor of 9)

9: 4 = 2 (1 left)
(9 is not divided by 4 without a remainder, so 4 is not a divisor of 9)

9:5 = 1 (4 left)
(9 is not divided by 5 without a remainder, so 5 is not a divisor of 9)

9: 6 = 1 (3 left)
(9 did not divide by 6 without a remainder, so 6 is not a divisor of 9)

9:7 = 1 (2 left)
(9 is not divided by 7 without a remainder, so 7 is not a divisor of 9)

9:8 = 1 (1 left)
(9 is not divided by 8 without a remainder, so 8 is not a divisor of 9)

9: 9 = 1
(9 divided by 9 without a remainder, so 9 is a divisor of 9)

Now write down the divisors of both numbers. The numbers highlighted in blue are the divisors. Let's write them out:

Having written out the divisors, you can immediately determine which one is the largest and most common.

By definition, the greatest common divisor of 12 and 9 is the number by which 12 and 9 are evenly divisible. The greatest and common divisor of the numbers 12 and 9 is the number 3

Both the number 12 and the number 9 are divisible by 3 without a remainder:

So gcd (12 and 9) = 3

The second way to find GCD

Now consider the second way to find the greatest common divisor. The essence of this method is to decompose both numbers into prime factors and multiply the common ones.

Example 1. Find GCD of numbers 24 and 18

First, let's factor both numbers into prime factors:

Now we multiply their common factors. In order not to get confused, the common factors can be underlined.

We look at the decomposition of the number 24. Its first factor is 2. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We underline both twos:

Again we look at the decomposition of the number 24. Its second factor is also 2. We are looking for the same factor in the decomposition of the number 18 and see that it is not there for the second time. Then we don't highlight anything.

The next two in the expansion of the number 24 is also missing in the expansion of the number 18.

We pass to the last factor in the decomposition of the number 24. This is the factor 3. We are looking for the same factor in the decomposition of the number 18 and we see that it is also there. We emphasize both threes:

So, the common factors of the numbers 24 and 18 are the factors 2 and 3. To get the GCD, these factors must be multiplied:

So gcd (24 and 18) = 6

The third way to find GCD

Now consider the third way to find the greatest common divisor. The essence of this method lies in the fact that the numbers to be searched for the greatest common divisor are decomposed into prime factors. Then, from the decomposition of the first number, factors that are not included in the decomposition of the second number are deleted. The remaining numbers in the first expansion are multiplied and get GCD.

For example, let's find the GCD for the numbers 28 and 16 in this way. First of all, we decompose these numbers into prime factors:

We got two expansions: and

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include seven. We will delete it from the first expansion:

Now we multiply the remaining factors and get the GCD:

The number 4 is the greatest common divisor of the numbers 28 and 16. Both of these numbers are divisible by 4 without a remainder:

Example 2 Find GCD of numbers 100 and 40

Factoring out the number 100

Factoring out the number 40

We got two expansions:

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include one five (there is only one five). We delete it from the first decomposition

Multiply the remaining numbers:

We got the answer 20. So the number 20 is the greatest common divisor of the numbers 100 and 40. These two numbers are divisible by 20 without a remainder:

GCD (100 and 40) = 20.

Example 3 Find the gcd of the numbers 72 and 128

Factoring out the number 72

Factoring out the number 128

2×2×2×2×2×2×2

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include two triplets (there are none at all). We delete them from the first expansion:

We got the answer 8. So the number 8 is the greatest common divisor of the numbers 72 and 128. These two numbers are divisible by 8 without a remainder:

GCD (72 and 128) = 8

Finding GCD for Multiple Numbers

The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found.

For example, let's find the GCD for the numbers 18, 24 and 36

Factoring the number 18

Factoring the number 24

Factoring the number 36

We got three expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all three numbers:

We see that the common factors for the numbers 18, 24 and 36 are factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 18, 24 and 36. These three numbers are divisible by 6 without a remainder:

GCD (18, 24 and 36) = 6

Example 2 Find gcd for numbers 12, 24, 36 and 42

Let's factorize each number. Then we find the product of the common factors of these numbers.

Factoring the number 12

Factoring the number 42

We got four expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all four numbers:

We see that the common factors for the numbers 12, 24, 36, and 42 are the factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 12, 24, 36 and 42. These numbers are divisible by 6 without a remainder:

gcd(12, 24, 36 and 42) = 6

From the previous lesson, we know that if some number is divided by another without a remainder, it is called a multiple of this number.

It turns out that a multiple can be common to several numbers. And now we will be interested in a multiple of two numbers, while it should be as small as possible.

Definition. Least common multiple (LCM) of numbers a and b- a and b a and number b.

Definition contains two variables a and b. Let's substitute any two numbers for these variables. For example, instead of a variable a substitute the number 9, and instead of the variable b let's substitute the number 12. Now let's try to read the definition:

Least common multiple (LCM) of numbers 9 and 12 - is the smallest number that is a multiple of 9 and 12 . In other words, it is such a small number that is divisible without a remainder by the number 9 and on the number 12 .

It is clear from the definition that the LCM is the smallest number that is divisible without a remainder by 9 and 12. This LCM is required to be found.

There are two ways to find the least common multiple (LCM). The first way is that you can write down the first multiples of two numbers, and then choose among these multiples such a number that will be common to both numbers and small. Let's apply this method.

First of all, let's find the first multiples for the number 9. To find the multiples for 9, you need to multiply this nine by the numbers from 1 to 9 in turn. The answers you get will be multiples of the number 9. So, let's start. Multiples will be highlighted in red:

Now we find multiples for the number 12. To do this, we multiply 12 by all the numbers 1 to 12 in turn.