Signs of divisibility 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.
This article is devoted to such a question as finding the greatest common divisor. First, we will explain what it is, and give a few examples, introduce the definitions of the greatest common divisor of 2, 3 or more numbers, after which we will dwell on the general properties of this concept and prove them.
Yandex.RTB R-A-339285-1
What are common divisors
To understand what the greatest common divisor is, we first formulate what a common divisor is for integers.
In the article on multiples and divisors, we said that an integer always has multiple divisors. Here we are interested in the divisors of a certain number of integers at once, especially common (identical) for all. Let us write down the main definition.
Definition 1
The common divisor of several integers will be a number that can be a divisor of each number from the specified set.
Example 1
Here are examples of such a divisor: the triple will be a common divisor for the numbers - 12 and 9, since the equalities 9 = 3 · 3 and − 12 = 3 · (− 4) are true. The numbers 3 and - 12 have other common divisors, such as 1 , - 1 and - 3 . Let's take another example. The four integers 3 , − 11 , − 8 and 19 will have two common divisors: 1 and - 1 .
Knowing the properties of divisibility, we can say that any integer can be divided by one and minus one, which means that any set of integers will already have at least two common divisors.
Also note that if we have a common divisor for several numbers b, then the same numbers can be divided by the opposite number, that is, by - b. In principle, we can only take positive divisors, then all common divisors will also be greater than 0 . This approach can also be used, but completely ignored negative numbers it does not follow.
What is the greatest common divisor (gcd)
According to the properties of divisibility, if b is a divisor of an integer a that is not equal to 0, then the modulus of b cannot be greater than the modulus of a, hence any number not equal to 0 has a finite number of divisors. This means that the number of common divisors of several integers, at least one of which differs from zero, will also be finite, and from their entire set we can always select the largest number (we have already talked about the concept of the largest and smallest integers, we advise you to repeat given material).
In further reasoning, we will assume that at least one of the set of numbers for which you need to find the greatest common divisor will be different from 0 . If they are all equal to 0 , then their divisor can be any integer, and since there are infinitely many of them, we cannot choose the largest. In other words, it is impossible to find the greatest common divisor for the set of numbers equal to 0 .
We pass to the formulation of the main definition.
Definition 2
The greatest common divisor of multiple numbers is the largest integer that divides all those numbers.
In writing, the greatest common divisor is most often denoted by the abbreviation GCD. For two numbers, it can be written as gcd (a, b) .
Example 2
What is an example of GCD for two integers? For example, for 6 and - 15 it would be 3 . Let's substantiate this. First, we write down all the divisors of six: ± 6, ± 3, ± 1, and then all the divisors of fifteen: ± 15, ± 5, ± 3 and ± 1. After that, we choose common ones: these are − 3 , − 1 , 1 and 3 . Of these, you need to choose the largest number. This will be 3 .
For three or more numbers, the definition of the greatest common divisor will be much the same.
Definition 3
The greatest common divisor of three or more numbers is the largest integer that divides all those numbers at the same time.
For numbers a 1 , a 2 , … , a n the divisor is conveniently denoted as GCD (a 1 , a 2 , … , a n) . The divisor value itself is written as GCD (a 1 , a 2 , … , a n) = b .
Example 3
Here are examples of the greatest common divisor of several integers: 12 , - 8 , 52 , 16 . It will be equal to four, which means we can write that gcd (12, - 8, 52, 16) = 4.
You can check the correctness of this statement by writing down all the divisors of these numbers and then choosing the largest of them.
In practice, there are often cases when the greatest common divisor is equal to one of the numbers. This happens when all other numbers can be divided by a given number (in the first paragraph of the article we gave the proof of this statement).
Example 4
So, the greatest common divisor of the numbers 60, 15 and - 45 is 15, since fifteen is divisible not only by 60 and - 45, but also by itself, and there is no greater divisor for all these numbers.
Coprime numbers are a special case. They are integers with a greatest common divisor of 1 .
Main properties of GCD and Euclid's algorithm
The greatest common divisor has some characteristic properties. We formulate them in the form of theorems and prove each of them.
Note that these properties are formulated for integers greater than zero, and we consider only positive divisors.
Definition 4
The numbers a and b have the greatest common divisor equal to gcd for b and a , i.e. gcd (a , b) = gcd (b , a) . Changing the places of numbers does not affect the final result.
This property follows from the very definition of GCD and does not need proof.
Definition 5
If the number a can be divided by the number b, then the set of common divisors of these two numbers will be similar to the set of divisors of the number b, that is, gcd (a, b) = b.
Let's prove this statement.
Proof 1
If the numbers a and b have common divisors, then any of them can be divided by them. At the same time, if a is a multiple of b, then any divisor of b will also be a divisor of a , since divisibility has such a property as transitivity. Hence, any divisor b will be common for the numbers a and b. This proves that if we can divide a by b , then the set of all divisors of both numbers coincides with the set of divisors of one number b . And since the greatest divisor of any number is the number itself, then the greatest common divisor of the numbers a and b will also be equal to b, i.e. gcd(a, b) = b. If a = b , then gcd (a , b) = gcd (a , a) = gcd (b , b) = a = b , e.g. gcd (132 , 132) = 132 .
Using this property, we can find the greatest common divisor of two numbers if one of them can be divided by the other. Such a divisor is equal to one of these two numbers by which the second number can be divided. For example, gcd (8, 24) = 8, because 24 is a multiple of eight.
Definition 6 Proof 2
Let's try to prove this property. We initially have the equality a = b q + c , and any common divisor of a and b will also divide c , which is explained by the corresponding divisibility property. Therefore, any common divisor of b and c will divide a . This means that the set of common divisors a and b will coincide with the set of divisors b and c, including the largest of them, which means that the equality gcd (a, b) = gcd (b, c) is true.
Definition 7
The following property is called the Euclid algorithm. With it, you can calculate the greatest common divisor of two numbers, as well as prove other properties of GCD.
Before formulating the property, we advise you to repeat the theorem that we proved in the article on division with a remainder. According to it, the divisible number a can be represented as b q + r, and here b is a divisor, q is some integer (it is also called an incomplete quotient), and r is a remainder that satisfies the condition 0 ≤ r ≤ b.
Let's say we have two integers greater than 0 for which the following equalities will be true:
a = b q 1 + r 1 , 0< r 1 < b b = r 1 · q 2 + r 2 , 0 < r 2 < r 1 r 1 = r 2 · q 3 + r 3 , 0 < r 3 < r 2 r 2 = r 3 · q 4 + r 4 , 0 < r 4 < r 3 ⋮ r k - 2 = r k - 1 · q k + r k , 0 < r k < r k - 1 r k - 1 = r k · q k + 1
These equalities end when r k + 1 becomes equal to 0 . This will happen for sure, since the sequence b > r 1 > r 2 > r 3 , … is a series of decreasing integers, which can include only a finite number of them. Hence, r k is the greatest common divisor of a and b , that is, r k = gcd (a , b) .
First of all, we need to prove that r k is a common divisor of the numbers a and b, and after that, that r k is not just a divisor, but the greatest common divisor of the two given numbers.
Let's look at the list of equalities above, from bottom to top. According to the last equality,
r k − 1 can be divided by r k . Based on this fact, as well as the previous proved property of the greatest common divisor, it can be argued that r k − 2 can be divided by r k , since
r k − 1 is divisible by r k and r k is divisible by r k .
The third equality from the bottom allows us to conclude that r k − 3 can be divided by r k , and so on. The second from the bottom is that b is divisible by r k , and the first is that a is divisible by r k . From all this we conclude that r k is a common divisor of a and b .
Now let's prove that r k = gcd (a , b) . What do I need to do? Show that any common divisor of a and b will divide r k . Let's denote it r 0 .
Let's look at the same list of equalities, but from top to bottom. Based on the previous property, we can conclude that r 1 is divisible by r 0 , which means that according to the second equality, r 2 is divisible by r 0 . We go down through all the equalities and from the last one we conclude that r k is divisible by r 0 . Therefore, r k = gcd (a , b) .
Having considered this property, we conclude that the set of common divisors of a and b is similar to the set of divisors of the gcd of these numbers. This statement, which is a consequence of Euclid's algorithm, will allow us to calculate all the common divisors of two given numbers.
Let's move on to other properties.
Definition 8
If a and b are integers not equal to 0, then there must be two other integers u 0 and v 0 for which the equality gcd (a , b) = a · u 0 + b · v 0 will be valid.
The equality given in the property statement is a linear representation of the greatest common divisor of a and b . It is called the Bezout ratio, and the numbers u 0 and v 0 are called the Bezout coefficients.
Proof 3
Let's prove this property. We write down the sequence of equalities according to the Euclid algorithm:
a = b q 1 + r 1 , 0< r 1 < b b = r 1 · q 2 + r 2 , 0 < r 2 < r 1 r 1 = r 2 · q 3 + r 3 , 0 < r 3 < r 2 r 2 = r 3 · q 4 + r 4 , 0 < r 4 < r 3 ⋮ r k - 2 = r k - 1 · q k + r k , 0 < r k < r k - 1 r k - 1 = r k · q k + 1
The first equality tells us that r 1 = a − b · q 1 . Denote 1 = s 1 and − q 1 = t 1 and rewrite this equality as r 1 = s 1 · a + t 1 · b . Here the numbers s 1 and t 1 will be integers. The second equality allows us to conclude that r 2 = b − r 1 q 2 = b − (s 1 a + t 1 b) q 2 = − s 1 q 2 a + (1 − t 1 q 2) b . Denote − s 1 q 2 = s 2 and 1 − t 1 q 2 = t 2 and rewrite the equality as r 2 = s 2 a + t 2 b , where s 2 and t 2 will also be integers. This is because the sum of integers, their product and difference are also integers. In exactly the same way, we obtain from the third equality r 3 = s 3 · a + t 3 · b , from the following r 4 = s 4 · a + t 4 · b, etc. Finally, we conclude that r k = s k a + t k b for integers s k and t k . Since r k \u003d GCD (a, b) , we denote s k \u003d u 0 and t k \u003d v 0. As a result, we can get a linear representation of GCD in the required form: GCD (a, b) \u003d a u 0 + b v 0.
Definition 9
gcd (m a, m b) = m gcd (a, b) for any natural value m.
Proof 4
This property can be justified as follows. Multiply by the number m both sides of each equality in the Euclid algorithm and we get that gcd (m a , m b) = m r k , and r k is gcd (a , b) . Hence, gcd (m a, m b) = m gcd (a, b) . It is this property of the greatest common divisor that is used when finding the GCD by the factorization method.
Definition 10
If numbers a and b have a common divisor p , then gcd (a: p , b: p) = gcd (a , b) : p . In the case when p = gcd (a , b) we get gcd (a: gcd (a , b) , b: gcd (a , b) = 1, therefore, the numbers a: gcd (a , b) and b: gcd (a , b) are coprime.
Since a = p (a: p) and b = p (b: p) , then, based on the previous property, we can create equalities of the form gcd (a , b) = gcd (p (a: p) , p · (b: p)) = p · GCD (a: p , b: p) , among which there will be a proof of this property. We use this assertion when we give common fractions to irreducible form.
Definition 11
The greatest common divisor a 1 , a 2 , … , a k will be the number d k , which can be found by successively calculating gcd (a 1 , a 2) = d 2 , gcd (d 2 , a 3) = d 3 , gcd (d 3 , a 4) = d 4 , … , GCD (d k - 1 , a k) = d k .
This property is useful for finding the greatest common divisor of three or more numbers. With it, you can reduce this action to operations with two numbers. Its basis is a corollary from Euclid's algorithm: if the set of common divisors a 1 , a 2 and a 3 coincides with the set d 2 and a 3 , then it also coincides with the divisors d 3 . The divisors of the numbers a 1 , a 2 , a 3 and a 4 will match the divisors of d 3 , which means they will also match the divisors of d 4 , and so on. In the end, we get that the common divisors of the numbers a 1 , a 2 , … , a k coincide with the divisors d k , and since this number itself will be the greatest divisor of the number d k , then gcd (a 1 , a 2 , … , a k) = d k .
That's all we would like to talk about the properties of the greatest common divisor.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
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 others can be divided by two. Therefore, only odd numbers can be prime.
Too many prime numbers complete list they don't exist. 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 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:
18 = 2×3×3
60 = 2×2×3×5
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 effective way finding 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.
- 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 common denominator when adding and subtracting fractions with different denominators 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.
- We write the multiples for each of the numbers in a line until there is a multiple that is the same for both numbers.
- 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.
LCM (24, 60) = 2 2 3 5 2
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 we 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
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 section "For study" you can download the table 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.
- decompose the divisors of numbers into prime factors;
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:
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 Special attention Let's take a look at the 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 and more numbers, and also pay attention to the calculation of the LCM of negative numbers.
Page navigation.
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: GCD(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: GCM(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 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 expansion of the number 75, we add the missing factors 2 and 7 from the expansion 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 expansion of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the expansion 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 .
www.cleverstudents.ru
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. essence this method is to factor 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 - this is smallest number, which is a multiple 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.
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 denote in the record capital letter TO.
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 different quantity multipliers.
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
So, the product of prime factors more and factors of the second number, which are not included in the expansion of the larger one, 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.
Loading...