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.
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 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 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.
- 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.
- 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 "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.
- 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 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: 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 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 decomposition:
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 found 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.
Consider two ways to find the greatest common divisor.
Finding by Factoring
The first way is to find the greatest common divisor by factoring given numbers into prime factors.
To find the GCD of several numbers, it is enough to decompose them into prime factors and multiply among themselves those of them that are common to all given numbers.
Example 1 Let's find GCD (84, 90).
We decompose the numbers 84 and 90 into prime factors:
So, we have underlined all the common prime factors, it remains to multiply them among themselves: 1 2 3 = 6.
So gcd(84, 90) = 6.
Example 2 Let's find GCD (15, 28).
We decompose 15 and 28 into prime factors:
The numbers 15 and 28 are coprime because their greatest common divisor is one.
gcd (15, 28) = 1.
Euclid's algorithm
The second method (otherwise called the Euclid method) is to find the GCD by successive division.
First, we will look at this method as applied to only two given numbers, and then we will figure out how to apply it to three or more numbers.
If the larger of two given numbers is divisible by the smaller, then the number that is smaller will be their greatest common divisor.
Example 1 Take two numbers 27 and 9. Since 27 is divisible by 9 and 9 is divisible by 9, then 9 is a common divisor of the numbers 27 and 9. This divisor is also the largest, because 9 cannot be divisible by any number, greater than 9. Therefore, gcd (27, 9) = 9.
In other cases, to find the greatest common divisor of two numbers, the following procedure is used:
- Of the two given numbers, the larger number is divided by the smaller one.
- Then, the smaller number is divided by the remainder resulting from dividing the larger number by the smaller one.
- Further, the first remainder is divided by the second remainder, which is obtained by dividing the smaller number by the first remainder.
- The second remainder is divided by the third, which is obtained by dividing the first remainder by the second, and so on.
- Thus, the division continues until the remainder is zero. The last divisor will be the greatest common divisor.
Example 2 Let's find the greatest common divisor of numbers 140 and 96:
1) 140: 96 = 1 (remainder 44)
2) 96: 44 = 2 (remainder 8)
3) 44: 8 = 5 (remainder 4)
The last divisor is 4, which means gcd(140, 96) = 4.
Sequential division can also be written in a column:
To find the greatest common divisor of three or more given numbers, use the following procedure:
- First, find the greatest common divisor of any two numbers from multiple datasets.
- Then we find the GCD of the found divisor and some third given number.
- Then we find the GCD of the last found divisor and the fourth given number, and so on.
Example 3 Let's find the greatest common divisor of the numbers 140, 96 and 48. We have already found the GCD of the numbers 140 and 96 in the previous example (this is the number 4). It remains to find the greatest common divisor of the number 4 and the third given number - 48:
48 is divisible by 4 without a remainder. So gcd(140, 96, 48) = 4.
Remember!
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.
For example:
- 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.
Remember!
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 a remainder.
Remember!
Greatest Common Divisor(GCD) of two given numbers "a" and "b" - this is the largest number by which both numbers "a" and "b" are divided without a remainder.
Briefly, the greatest common divisor of the numbers "a" and "b" is written as follows:
gcd (a; b) .
Example: gcd (12; 36) = 12 .
The divisors of numbers in the solution record are denoted by a capital letter "D".
D(7) = (1, 7)
D(9) = (1, 9)
gcd (7; 9) = 1
The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.
Remember!
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 2 764 = 2 2 2 2 2 2
- We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4Answer: 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”.
Now and in what follows, we will assume that at least one of these numbers is different from zero. If all given numbers are equal to zero, then their common divisor is any integer, and since there are infinitely many integers, we cannot talk about the largest of them. Therefore, one cannot speak of the greatest common divisor of numbers, each of which is equal to zero.
Now we can give finding the greatest common divisor two numbers.
Definition.
Greatest Common Divisor of two integers is the largest integer that divides the two given integers.
The abbreviation GCD is often used to shorten the greatest common divisor - Greatest Common Divisor. Also, the greatest common divisor of two numbers a and b is often denoted as gcd(a, b) .
Let's bring Greatest Common Divisor (gcd) example two integers. The greatest common divisor of 6 and −15 is 3 . Let's substantiate this. Let's write down all the divisors of the number six: ±6, ±3, ±1, and the divisors of the number −15 are the numbers ±15, ±5, ±3 and ±1. Now you can find all the common divisors of the numbers 6 and −15, these are the numbers −3, −1, 1 and 3. Since −3<−1<1<3 , то 3 – это наибольший общий делитель чисел 6 и −15 . То есть, НОД(6, −15)=3 .
The definition of the greatest common divisor of three or more integers is similar to the definition of gcd of two numbers.
Definition.
Greatest Common Divisor three or more integers is the largest integer that simultaneously divides all given numbers.
The greatest common divisor of n integers a 1 , a 2 , …, a n we will denote as gcd(a 1 , a 2 , …, a n) . If the value b of the greatest common divisor of these numbers is found, then we can write GCD(a 1 , a 2 , …, a n)=b.
As an example, given the gcd of four integers −8 , 52 , 16 and −12 , it is equal to 4 , that is, gcd(−8, 52, 16, −12)=4 . This can be checked by writing down all the divisors of the given numbers, selecting the common divisors from them, and determining the greatest common divisor.
Note that the greatest common divisor of integers can be equal to one of these numbers. This statement is true if all given numbers are divisible by one of them (the proof is given in the next paragraph of this article). For example, gcd(15, 60, −45)=15 . This is true because 15 divides 15 , 60 , and −45 , and there is no common divisor of 15 , 60 , and −45 that is greater than 15 .
Of particular interest are the so-called relatively prime numbers, - such integers, the greatest common divisor of which is equal to one.
Greatest Common Divisor Properties, Euclid's Algorithm
The greatest common divisor has a number of characteristic results, in other words, a number of properties. We will now list the main properties of the greatest common divisor (gcd), we will formulate them in the form of theorems and immediately give proofs.
We will formulate all the properties of the greatest common divisor for positive integers, while we will consider only positive divisors of these numbers.
The greatest common divisor of a and b is equal to the greatest common divisor of b and a , that is, gcd(a, b)=gcd(a, b) .
This GCD property follows directly from the definition of the greatest common divisor.
If a is divisible by b , then the set of common divisors of a and b is the same as the set of divisors of b , in particular gcd(a, b)=b .
Proof.
Any common divisor of the numbers a and b is a divisor of each of these numbers, including the number b. On the other hand, since a is a multiple of b, then any divisor of the number b is also a divisor of the number a due to the fact that divisibility has the property of transitivity, therefore, any divisor of the number b is a common divisor of the numbers a and b. This proves that if a is divisible by b, then the set of divisors of the numbers a and b coincides with the set of divisors of one number b. And since the greatest divisor of the number b is the number b itself, then the greatest common divisor of the numbers a and b is also equal to b , that is, gcd(a, b)=b .
In particular, if the numbers a and b are equal, then gcd(a, b)=gcd(a, a)=gcd(b, b)=a=b. For example, gcd(132, 132)=132 .
The proven greatest divisor property allows us to find the gcd of two numbers when one of them is divisible by the other. In this case, the GCD is equal to one of these numbers, by which another number is divisible. For example, gcd(8, 24)=8 since 24 is a multiple of eight.
If a=b q+c , where a , b , c and q are integers, then the set of common divisors of the numbers a and b coincides with the set of common divisors of the numbers b and c , in particular, gcd(a, b)=gcd (b, c) .
Let us justify this property of the GCD.
Since the equality a=b·q+c holds, then any common divisor of numbers a and b also divides c (this follows from the properties of divisibility). For the same reason, every common divisor of b and c divides a . Therefore, the set of common divisors of the numbers a and b is the same as the set of common divisors of the numbers b and c. In particular, the largest of these common divisors must also match, that is, the following equality must be valid gcd(a, b)=gcd(b, c) .
Now we formulate and prove a theorem, which is Euclid's algorithm. Euclid's algorithm allows you to find the GCD of two numbers (see finding the GCD using the Euclid algorithm). Moreover, Euclid's algorithm will allow us to prove the following properties of the greatest common divisor.
Before giving the statement of the theorem, we recommend refreshing the memory of the theorem from the theory section, which states that the dividend a can be represented as b q + r, where b is a divisor, q is some integer called the partial quotient, and r is an integer that satisfies the condition, called the remainder.
So, let for two non-zero positive integers a and b, a series of equalities is true 
ending when r k+1 =0 (which is inevitable, since b>r 1 >r 2 >r 3 , … is a series of decreasing integers, and this series cannot contain more than a finite number of positive numbers), then r k – is the greatest common divisor of a and b , that is, r k =gcd(a, b) .
Proof.
Let us first prove that r k is a common divisor of numbers a and b , after which we will show that r k is not just a divisor, but the greatest common divisor of numbers a and b .
We will move along the written equalities from bottom to top. From the last equality, we can say that r k−1 is divisible by r k . Given this fact, as well as the previous GCD property, the penultimate equality r k−2 =r k−1 q k +r k allows us to assert that r k−2 is divisible by r k , since r k−1 is divisible by r k and r k is divisible by r k . By analogy, from the third equality from the bottom we conclude that r k−3 is divisible by r k . And so on. From the second equality we get that b is divisible by r k , and from the first equality we get that a is divisible by r k . Therefore, r k is a common divisor of a and b.
It remains to prove that r k =gcd(a, b) . For, it suffices to show that any common divisor of numbers a and b (we denote it by r 0 ) divides r k .
We will move along the initial equalities from top to bottom. By virtue of the previous property, it follows from the first equality that r 1 is divisible by r 0 . Then from the second equality we get that r 2 is divisible by r 0 . And so on. From the last equality we get that r k is divisible by r 0 . Thus, r k =gcd(a, b) .
It follows from the considered property of the greatest common divisor that the set of common divisors of the numbers a and b coincides with the set of divisors of the greatest common divisor of these numbers. This corollary from Euclid's algorithm allows us to find all common divisors of two numbers as divisors of the gcd of these numbers.
Let a and b be integers not simultaneously equal to zero, then there are such integers u 0 and v 0 , then the equality gcd(a, b)=a u 0 +b v 0 is true. The last equality is a linear representation of the greatest common divisor of the numbers a and b, this equality is called the Bezout ratio, and the numbers u 0 and v 0 are the Bezout coefficients.
Proof.
According to Euclid's algorithm, we can write the following equalities
From the first equality we have r 1 =a−b q 1 , and, denoting 1=s 1 and −q 1 =t 1 , this equality takes the form r 1 =s 1 a+t 1 b , and the numbers s 1 and t 1 are integers. Then from the second equality we obtain 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. Denoting −s 1 q 2 =s 2 and 1−t 1 q 2 =t 2 , the last equality can be written as r 2 =s 2 a+t 2 b , and s 2 and t 2 are integers (because the sum, difference and product of integers is an integer). Similarly, from the third equality we get r 3 =s 3 ·a+t 3 ·b, from the fourth r 4 =s 4 ·a+t 4 ·b, and so on. Finally, r k =s k ·a+t k ·b , where s k and t k are integers. Since r k =gcd(a, b) , and denoting s k =u 0 and t k =v 0 , we obtain a linear representation of the gcd of the required form: gcd(a, b)=a u 0 +b v 0 .
If m is any natural number, then gcd(m a, m b)=m gcd(a, b).
The rationale for this property of the greatest common divisor is as follows. If we multiply by m both sides of each of the equalities of the Euclid algorithm, we get that gcd(m a, m b)=m r k , and r k is gcd(a, b) . Consequently, gcd(m a, m b)=m gcd(a, b).
This property of the greatest common divisor is the basis for the method of finding GCD using prime factorization.
Let p be any common divisor of numbers a and b , then gcd(a:p, b:p)=gcd(a, b):p, in particular, if p=gcd(a, b) we have gcd(a:gcd(a, b), b:gcd(a, b))=1, that is, the numbers a:gcd(a, b) and b:gcd(a, b) are coprime.
Since a=p (a:p) and b=p (b:p) , and due to the previous property, we can write a chain of equalities of the form gcd(a, b)=gcd(p (a:p), p (b:p))= p·gcd(a:p, b:p) , whence the equality to be proved follows.
The greatest common divisor property just proved underlies .
Now let's voice the GCD property, which reduces the problem of finding the greatest common divisor of three or more numbers to successively finding the GCD of two numbers.
The greatest common divisor of numbers a 1 , a 2 , ..., a k is equal to the number d k , which is found in the sequential calculation of 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 .
The proof is based on a corollary from Euclid's algorithm. The common divisors of the numbers a 1 and a 2 are the same as the divisors of d 2 . Then the common divisors of the numbers a 1 , a 2 and a 3 coincide with the common divisors of the numbers d 2 and a 3 , therefore, they coincide with the divisors of d 3 . The common divisors of the numbers a 1 , a 2 , a 3 and a 4 are the same as the common divisors of d 3 and a 4 , hence the same as the divisors of d 4 . And so on. Finally, the common divisors of the numbers a 1 , a 2 , …, a k coincide with the divisors of d k . And since the greatest divisor of the number d k is the number d k itself, then GCD(a 1 , a 2 , …, a k)=d k.
This concludes the review of the main properties of the greatest common divisor.
Bibliography.
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- Mikhelovich Sh.Kh. Number theory.
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