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 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.
Example.
Find the least common multiple of the two numbers 126 and 70 .
Solution.
In this example a=126 , b=70 . Let us use the relationship between LCM and GCD 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: GCM(126, 70)= 126 70:14=630 .
Answer:
LCM(126, 70)=630 .
Example.
What is LCM(68, 34) ?
Solution.
Because 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34: LCM(68, 34)= 68 34:34=68 .
Answer:
LCM(68, 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 the numbers 75 and 210, that is, LCM(75, 210)= 2 3 5 5 7=1 050.
Example.
After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.
Solution.
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 . In this way, LCM(441, 700)=2 2 3 3 5 5 7 7=44 100.
Answer:
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) .
Example.
Find the least common multiple of 84 and 648.
Solution.
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.
Answer:
LCM(84, 648)=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.
Theorem.
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.
Example.
Find the LCM of the four numbers 140 , 9 , 54 and 250 .
Solution.
In this example a 1 =140 , a 2 =9 , a 3 =54 , a 4 =250 .
First we find m 2 \u003d LCM (a 1, a 2) \u003d 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: LCM(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .
Now we find m 3 \u003d LCM (m 2, a 3) \u003d 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.
Left to find m 4 \u003d LCM (m 3, a 4) \u003d 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 , whence gcd(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.
Answer:
LCM(140, 9, 54, 250)=94,500.
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.
Example.
Find the least common multiple of five numbers 84 , 6 , 48 , 7 , 143 .
Solution.
First, we obtain the expansions of these numbers into prime factors: 84=2 2 3 7 , 6=2 3 , 48=2 2 2 2 3 , 7 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 .
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 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 check out finding the least common multiple (LCM) in the following way. 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.
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: GCM(a, b). Consider examples of finding the LCM according to the above formula.
Find the least common multiple of the two numbers 126 and 70 .
In this example a=126 , b=70 . Let's use the link of LCM with GCD, which is expressed by the formula LCM(a, b)=a b: GCM(a, b) . That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers according to the written formula.
Find gcd(126, 70) using Euclid's algorithm: 126=70 1+56 , 70=56 1+14 , 56=14 4 , hence gcd(126, 70)=14 .
Now we find the required least common multiple: LCM(126, 70)=126 70:GCD(126, 70)= 126 70:14=630 .
What is LCM(68, 34) ?
Since 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34:GCD(68, 34)= 68 34:34=68 .
Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b , then the least common multiple of these numbers is a .
Finding the LCM by Factoring Numbers into Prime Factors
Another way to find the least common multiple is based on factoring numbers into prime factors. If we make a product of all prime factors of these numbers, after which we exclude from this product all common prime factors that are present in the expansions of these numbers, then the resulting product will be equal to the least common multiple of these numbers.
The announced rule for finding the LCM follows from the equality LCM(a, b)=a b: GCD(a, b) . Indeed, the product of the numbers a and b is equal to the product of all the factors involved in the expansions of the numbers a and b. In turn, gcd(a, b) is equal to the product of all prime factors that are simultaneously present in the expansions of the numbers a and b (which is described in the section on finding the gcd using the decomposition of numbers into prime factors).
Let's take an example. Let we know that 75=3 5 5 and 210=2 3 5 7 . Compose the product of all factors of these expansions: 2 3 3 5 5 5 7 . Now we exclude from this product all the factors that are present both in the expansion of the number 75 and in the expansion of the number 210 (such factors are 3 and 5), then the product will take the form 2 3 5 5 7 . The value of this product is equal to the least common multiple of 75 and 210 , that is, LCM(75, 210)= 2 3 5 5 7=1 050 .
After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.
Let's decompose the numbers 441 and 700 into prime factors: 
We get 441=3 3 7 7 and 700=2 2 5 5 7 .
Now let's make a product of all the factors involved in the expansions of these numbers: 2 2 3 3 5 5 7 7 7 . Let us exclude from this product all the factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2 2 3 3 5 5 7 7 . So LCM(441, 700)=2 2 3 3 5 5 7 7=44 100 .
LCM(441, 700)= 44 100 .
The rule for finding the LCM using the decomposition of numbers into prime factors can be formulated a little differently. If we add the missing factors from the expansion of the number b to the factors from the expansion of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.
For example, let's take all the same numbers 75 and 210, their expansions into prime factors are as follows: 75=3 5 5 and 210=2 3 5 7 . To the factors 3, 5 and 5 from the decomposition of the number 75, we add the missing factors 2 and 7 from the decomposition of the number 210, we get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .
Find the least common multiple of 84 and 648.
We first obtain the decomposition of the numbers 84 and 648 into prime factors. They look like 84=2 2 3 7 and 648=2 2 2 3 3 3 3 . To the factors 2 , 2 , 3 and 7 from the decomposition of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the decomposition of the number 648 , we get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 84 and 648 is 4,536.
Finding the LCM of three or more numbers
The least common multiple of three or more numbers can be found by successively finding the LCM of two numbers. Recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.
Let positive integers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k =LCM(m k−1 , a k) .
Consider the application of this theorem on the example of finding the least common multiple of four numbers.
Find the LCM of the four numbers 140 , 9 , 54 and 250 .
First we find m 2 = LCM (a 1 , a 2) = LCM (140, 9) . To do this, using the Euclidean algorithm, we determine gcd(140, 9) , we have 140=9 15+5 , 9=5 1+4 , 5=4 1+1 , 4=1 4 , therefore, gcd(140, 9)=1 , whence LCM(140, 9)=140 9: GCD(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .
Now we find m 3 = LCM (m 2 , a 3) = LCM (1 260, 54) . Let's calculate it through gcd(1 260, 54) , which is also determined by the Euclid algorithm: 1 260=54 23+18 , 54=18 3 . Then gcd(1 260, 54)=18 , whence LCM(1 260, 54)= 1 260 54:gcd(1 260, 54)= 1 260 54:18=3 780 . That is, m 3 \u003d 3 780.
It remains to find m 4 = LCM (m 3 , a 4) = LCM (3 780, 250) . To do this, we find GCD(3 780, 250) using the Euclid algorithm: 3 780=250 15+30 , 250=30 8+10 , 30=10 3 . Therefore, gcd(3 780, 250)=10 , hence LCM(3 780, 250)= 3 780 250:gcd(3 780, 250)= 3 780 250:10=94 500 . That is, m 4 \u003d 94 500.
So the least common multiple of the original four numbers is 94,500.
LCM(140, 9, 54, 250)=94500 .
In many cases, the least common multiple of three or more numbers is conveniently found using prime factorizations of given numbers. In this case, the following rule should be followed. The least common multiple of several numbers is equal to the product, which is composed as follows: the missing factors from the expansion of the second number are added to all the factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the obtained factors, and so on.
Consider an example of finding the least common multiple using the decomposition of numbers into prime factors.
Find the least common multiple of five numbers 84 , 6 , 48 , 7 , 143 .
First, we obtain decompositions of these numbers into prime factors: 84=2 2 3 7 , 6=2 3 , 48=2 2 2 2 3 , 7 (7 is a prime number, it coincides with its decomposition into prime factors) and 143=11 13 .
To find the LCM of these numbers, to the factors of the first number 84 (they are 2 , 2 , 3 and 7) you need to add the missing factors from the expansion of the second number 6 . The expansion of the number 6 does not contain missing factors, since both 2 and 3 are already present in the expansion of the first number 84 . Further to the factors 2 , 2 , 3 and 7 we add the missing factors 2 and 2 from the expansion of the third number 48 , we get a set of factors 2 , 2 , 2 , 2 , 3 and 7 . There is no need to add factors to this set in the next step, since 7 is already contained in it. Finally, to the factors 2 , 2 , 2 , 2 , 3 and 7 we add the missing factors 11 and 13 from the expansion of the number 143 . We get the product 2 2 2 2 3 7 11 13 , which is equal to 48 048 .
Therefore, LCM(84, 6, 48, 7, 143)=48048 .
LCM(84, 6, 48, 7, 143)=48048 .
Finding the Least Common Multiple of Negative Numbers
Sometimes there are tasks in which you need to find the least common multiple of numbers, among which one, several or all numbers are negative. In these cases, all negative numbers must be replaced by their opposite numbers, after which the LCM of positive numbers should be found. This is the way to find the LCM of negative numbers. For example, LCM(54, −34)=LCM(54, 34) and LCM(−622, −46, −54, −888)= LCM(622, 46, 54, 888) .
We can do this because the set of multiples of a is the same as the set of multiples of −a (a and −a are opposite numbers). Indeed, let b be some multiple of a , then b is divisible by a , and the concept of divisibility asserts the existence of such an integer q that b=a q . But the equality b=(−a)·(−q) will also be true, which, by virtue of the same concept of divisibility, means that b is divisible by −a , that is, b is a multiple of −a . The converse statement is also true: if b is some multiple of −a , then b is also a multiple of a .
Find the least common multiple of the negative numbers −145 and −45.
Let's replace the negative numbers −145 and −45 with their opposite numbers 145 and 45 . We have LCM(−145, −45)=LCM(145, 45) . Having determined gcd(145, 45)=5 (for example, using the Euclid algorithm), we calculate LCM(145, 45)=145 45:gcd(145, 45)= 145 45:5=1 305 . Thus, the least common multiple of the negative integers −145 and −45 is 1,305 .
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We continue to study division. AT this lesson We will consider 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 dividing 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.
How to find LCM (least common multiple)
The common multiple of two integers is the integer that is evenly divisible by both given numbers without remainder.The least common multiple of two integers is the smallest of all integers that is divisible evenly and without remainder by both given numbers.
Method 1. You can find the LCM, in turn, for each of the given numbers, writing out in ascending order all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.
Example for numbers 6 and 9.
We multiply the number 6, sequentially, by 1, 2, 3, 4, 5.
We get: 6, 12, 18
, 24, 30
We multiply the number 9, sequentially, by 1, 2, 3, 4, 5.
We get: 9, 18
, 27, 36, 45
As you can see, the LCM for the numbers 6 and 9 will be 18.
This method is convenient when both numbers are small and it is easy to multiply them by a sequence of integers. However, there are cases when you need to find the LCM for two-digit or three-digit numbers, and also when there are three or even more initial numbers.
Method 2. You can find the LCM by decomposing the original numbers into prime factors.
After decomposition, it is necessary to cross out the same numbers from the resulting series of prime factors. The remaining numbers of the first number will be the factor for the second, and the remaining numbers of the second number will be the factor for the first.
Example for the number 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing out multiples of these numbers in a row. To do this, we decompose 75 and 60 into prime factors:
75 = 3
* 5
* 5, and
60 = 2 * 2 * 3
* 5
.
As you can see, the factors 3 and 5 occur in both rows. Mentally we "cross out" them.
Let's write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we left the number 5, and when decomposing the number 60, we left 2 * 2
So, to determine the LCM for the numbers 75 and 60, we need to multiply the remaining numbers from the expansion of 75 (this is 5) by 60, and the numbers remaining from the expansion of the number 60 (this is 2 * 2) multiply by 75. That is, for ease of understanding , we say that we multiply "crosswise".
75 * 2 * 2 = 300
60 * 5 = 300
This is how we found the LCM for the numbers 60 and 75. This is the number 300.
Example. Determine LCM for numbers 12, 16, 24
AT this case, our actions will be somewhat more complicated. But, first, as always, we decompose all numbers into prime factors
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
To correctly determine the LCM, we select the smallest of all numbers (this is the number 12) and successively go through its factors, crossing them out if at least one of the other rows of numbers has the same factor that has not yet been crossed out.
Step 1 . We see that 2 * 2 occurs in all series of numbers. We cross them out.
12 = 2
* 2
* 3
16 = 2
* 2
* 2 * 2
24 = 2
* 2
* 2 * 3
Step 2. In the prime factors of the number 12, only the number 3 remains. But it is present in the prime factors of the number 24. We cross out the number 3 from both rows, while no action is expected for the number 16.
12 = 2
* 2
* 3
16 = 2
* 2
* 2 * 2
24 = 2
* 2
* 2 * 3
As you can see, when decomposing the number 12, we "crossed out" all the numbers. So the finding of the NOC is completed. It remains only to calculate its value.
For the number 12, we take the remaining factors from the number 16 (the closest in ascending order)
12 * 2 * 2 = 48
This is the NOC
As you can see, in this case, finding the LCM was somewhat more difficult, but when you need to find it for three or more numbers, this method allows you to do it faster. However, both ways of finding the LCM are correct.
Greatest Common Divisor
Definition 2
If a natural number a is divisible by a natural number $b$, then $b$ is called a divisor of $a$, and the number $a$ is called a multiple of $b$.
Let $a$ and $b$ be natural numbers. The number $c$ is called a common divisor for both $a$ and $b$.
The set of common divisors of the numbers $a$ and $b$ is finite, since none of these divisors can be greater than $a$. This means that among these divisors there is the largest one, which is called the greatest common divisor of the numbers $a$ and $b$, and the notation is used to denote it:
$gcd \ (a;b) \ or \ D \ (a;b)$
To find the greatest common divisor of two numbers:
- Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.
Example 1
Find the gcd of the numbers $121$ and $132.$
$242=2\cdot 11\cdot 11$
$132=2\cdot 2\cdot 3\cdot 11$
Choose the numbers that are included in the expansion of these numbers
$242=2\cdot 11\cdot 11$
$132=2\cdot 2\cdot 3\cdot 11$
Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.
$gcd=2\cdot 11=22$
Example 2
Find the GCD of monomials $63$ and $81$.
We will find according to the presented algorithm. For this:
Let's decompose numbers into prime factors
$63=3\cdot 3\cdot 7$
$81=3\cdot 3\cdot 3\cdot 3$
We select the numbers that are included in the expansion of these numbers
$63=3\cdot 3\cdot 7$
$81=3\cdot 3\cdot 3\cdot 3$
Let's find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.
$gcd=3\cdot 3=9$
You can find the GCD of two numbers in another way, using the set of divisors of numbers.
Example 3
Find the gcd of the numbers $48$ and $60$.
Solution:
Find the set of divisors of $48$: $\left\((\rm 1,2,3.4.6,8,12,16,24,48)\right\)$
Now let's find the set of divisors of $60$:$\ \left\((\rm 1,2,3,4,5,6,10,12,15,20,30,60)\right\)$
Let's find the intersection of these sets: $\left\((\rm 1,2,3,4,6,12)\right\)$ - this set will determine the set of common divisors of the numbers $48$ and $60$. The largest element in this set will be the number $12$. So the greatest common divisor of $48$ and $60$ is $12$.
Definition of NOC
Definition 3
common multiple of natural numbers$a$ and $b$ is a natural number that is a multiple of both $a$ and $b$.
Common multiples of numbers are numbers that are divisible by the original without a remainder. For example, for the numbers $25$ and $50$, the common multiples will be the numbers $50,100,150,200$, etc.
The least common multiple will be called the least common multiple and denoted by LCM$(a;b)$ or K$(a;b).$
To find the LCM of two numbers, you need:
- Decompose numbers into prime factors
- Write out the factors that are part of the first number and add to them the factors that are part of the second and do not go to the first
Example 4
Find the LCM of the numbers $99$ and $77$.
We will find according to the presented algorithm. For this
Decompose numbers into prime factors
$99=3\cdot 3\cdot 11$
Write down the factors included in the first
add to them factors that are part of the second and do not go to the first
Find the product of the numbers found in step 2. The resulting number will be the desired least common multiple
$LCC=3\cdot 3\cdot 11\cdot 7=693$
Compiling lists of divisors of numbers is often very time consuming. There is a way to find GCD called Euclid's algorithm.
Statements on which Euclid's algorithm is based:
If $a$ and $b$ are natural numbers, and $a\vdots b$, then $D(a;b)=b$
If $a$ and $b$ are natural numbers such that $b
Using $D(a;b)= D(a-b;b)$, we can successively decrease the numbers under consideration until we reach a pair of numbers such that one of them is divisible by the other. Then the smaller of these numbers will be the desired greatest common divisor for the numbers $a$ and $b$.
Properties of GCD and LCM
- Any common multiple of $a$ and $b$ is divisible by K$(a;b)$
- If $a\vdots b$ , then K$(a;b)=a$
If K$(a;b)=k$ and $m$-natural number, then K$(am;bm)=km$
If $d$ is a common divisor for $a$ and $b$, then K($\frac(a)(d);\frac(b)(d)$)=$\ \frac(k)(d) $
If $a\vdots c$ and $b\vdots c$ , then $\frac(ab)(c)$ is a common multiple of $a$ and $b$
For any natural numbers $a$ and $b$ the equality
$D(a;b)\cdot K(a;b)=ab$
Any common divisor of $a$ and $b$ is a divisor of $D(a;b)$
Lancinova Aisa
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Tasks for GCD and LCM of numbers The work of a 6th grade student of the MKOU "Kamyshovskaya OOSh" Lantsinova Aisa Supervisor Goryaeva Zoya Erdnigoryaevna, teacher of mathematics p. Kamyshovo, 2013
An example of finding the GCD of the numbers 50, 75 and 325. 1) Let's decompose the numbers 50, 75 and 325 into prime factors. 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙ 13 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙13 divide without a remainder the numbers a and b are called the greatest common divisor of these numbers.
An example of finding the LCM of the numbers 72, 99 and 117. 1) Let us factorize the numbers 72, 99 and 117. Write out the factors included in the expansion of one of the numbers 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 and add to them the missing factors of the remaining numbers. 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 11 ∙ 13 3) Find the product of the resulting factors. 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 11 ∙ 13= 10296 Answer: LCM (72, 99 and 117) = 10296 The least common multiple of natural numbers a and b is the smallest natural number that is a multiple of a and b.
A sheet of cardboard has the shape of a rectangle, the length of which is 48 cm and the width is 40 cm. This sheet must be cut without waste into equal squares. What are the largest squares that can be obtained from this sheet and how many? Solution: 1) S = a ∙ b is the area of the rectangle. S \u003d 48 ∙ 40 \u003d 1960 cm². is the area of the cardboard. 2) a - the side of the square 48: a - the number of squares that can be laid along the length of the cardboard. 40: a - the number of squares that can be laid across the width of the cardboard. 3) GCD (40 and 48) \u003d 8 (cm) - the side of the square. 4) S \u003d a² - the area of \u200b\u200bone square. S \u003d 8² \u003d 64 (cm².) - the area of \u200b\u200bone square. 5) 1960: 64 = 30 (number of squares). Answer: 30 squares with a side of 8 cm each. Tasks for GCD
The fireplace in the room must be laid out with finishing tiles in the shape of a square. How many tiles will be needed for a 195 ͯ 156 cm fireplace and what are largest dimensions tiles? Solution: 1) S = 196 ͯ 156 = 30420 (cm ²) - S of the fireplace surface. 2) GCD (195 and 156) = 39 (cm) - side of the tile. 3) S = a² = 39² = 1521 (cm²) - area of 1 tile. 4) 30420: = 20 (pieces). Answer: 20 tiles measuring 39 ͯ 39 (cm). Tasks for GCD
A garden plot measuring 54 ͯ 48 m around the perimeter must be fenced off, for this, concrete pillars must be placed at regular intervals. How many poles must be brought for the site, and at what maximum distance from each other will the poles stand? Solution: 1) P = 2(a + b) – site perimeter. P \u003d 2 (54 + 48) \u003d 204 m. 2) GCD (54 and 48) \u003d 6 (m) - the distance between the pillars. 3) 204: 6 = 34 (pillars). Answer: 34 pillars, at a distance of 6 m. Tasks for GCD
Out of 210 burgundy, 126 white, 294 red roses, bouquets were collected, and in each bouquet the number of roses of the same color is equal. What is the largest number of bouquets made from these roses and how many roses of each color are in one bouquet? Solution: 1) GCD (210, 126 and 294) = 42 (bouquets). 2) 210: 42 = 5 (burgundy roses). 3) 126: 42 = 3 (white roses). 4) 294: 42 = 7 (red roses). Answer: 42 bouquets: 5 burgundy, 3 white, 7 red roses in each bouquet. Tasks for GCD
Tanya and Masha bought the same number of mailboxes. Tanya paid 90 rubles, and Masha paid 5 rubles. more. How much does one set cost? How many sets did each buy? Solution: 1) Masha paid 90 + 5 = 95 (rubles). 2) GCD (90 and 95) = 5 (rubles) - the price of 1 set. 3) 980: 5 = 18 (sets) - bought by Tanya. 4) 95: 5 = 19 (sets) - Masha bought. Answer: 5 rubles, 18 sets, 19 sets. Tasks for GCD
Three tourist boat trips start in the port city, the first of which lasts 15 days, the second - 20 and the third - 12 days. Returning to the port, the ships on the same day again go on a voyage. Motor ships left the port on all three routes today. In how many days will they sail together for the first time? How many trips will each ship make? Solution: 1) NOC (15.20 and 12) = 60 (days) - meeting time. 2) 60: 15 = 4 (voyages) - 1 ship. 3) 60: 20 = 3 (voyages) - 2 motor ship. 4) 60: 12 = 5 (voyages) - 3 motor ship. Answer: 60 days, 4 flights, 3 flights, 5 flights. Tasks for the NOC
Masha bought eggs for the Bear in the store. On the way to the forest, she realized that the number of eggs is divisible by 2,3,5,10 and 15. How many eggs did Masha buy? Solution: LCM (2;3;5;10;15) = 30 (eggs) Answer: Masha bought 30 eggs. Tasks for the NOC
It is required to make a box with a square bottom for stacking boxes measuring 16 ͯ 20 cm. What should be the shortest side of the square bottom to fit the boxes tightly into the box? Solution: 1) NOC (16 and 20) = 80 (boxes). 2) S = a ∙ b is the area of 1 box. S \u003d 16 ∙ 20 \u003d 320 (cm ²) - the area of the bottom of 1 box. 3) 320 ∙ 80 = 25600 (cm ²) - square bottom area. 4) S \u003d a² \u003d a ∙ a 25600 \u003d 160 ∙ 160 - the dimensions of the box. Answer: 160 cm is the side of the square bottom. Tasks for the NOC
Along the road from point K there are power poles every 45 m. It was decided to replace these poles with others, placing them at a distance of 60 m from each other. How many poles were there and how many will they stand? Solution: 1) NOK (45 and 60) = 180. 2) 180: 45 = 4 - there were pillars. 3) 180: 60 = 3 - there were pillars. Answer: 4 pillars, 3 pillars. Tasks for the NOC
How many soldiers are marching on the parade ground if they march in formation of 12 people in a line and change into a column of 18 people in a line? Solution: 1) NOC (12 and 18) = 36 (people) - marching. Answer: 36 people. Tasks for the NOC