To understand how to calculate the LCM, you must first determine the meaning of the term “multiple”.
A multiple of A is a natural number that is divisible by A without a remainder. Thus, numbers that are multiples of 5 can be considered 15, 20, 25, and so on.
There can be a limited number of divisors of a particular number, but there are an infinite number of multiples.
A common multiple of natural numbers is a number that is divisible by them without leaving a remainder.
How to find the least common multiple of numbers
The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is divisible by all these numbers.
To find the LOC, you can use several methods.
For small numbers, it is convenient to write down all the multiples of these numbers on a line until you find something common among them. Multiples are denoted by the capital letter K.
For example, multiples of 4 can be written like this:
K (4) = (8,12, 16, 20, 24, ...)
K (6) = (12, 18, 24, ...)
Thus, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This notation is done as follows:
LCM(4, 6) = 24
Now write down the common factors for both numbers. In our version it is two and five. However, in other cases this number can be one, two or three digits or even more. Next you need to work with degrees. Choose the smallest power for each factor. In the example it is two to the second power and five to the first.
Finally, you just need to multiply the resulting numbers. In our case, everything is extremely simple: two squared multiplied by five equals 20. Thus, the number 20 can be called the greatest common divisor for 60 and 80.
Video on the topic
note
Remember that a prime factor is a number that has only 2 divisors: one and the number itself.
Helpful advice
In addition to this method, you can also use the Euclidean algorithm. Its full description, presented in geometric form, can be found in Euclid's book "Elements".
Related article
Addition and subtraction of natural fractions is only possible if they have the same denominator. In order not to complicate the calculations when bringing them to a single denominator, find the least common divisor of the denominators and carry out the calculation.
You will need
- - ability to factor numbers into prime factors;
- - ability to perform operations with fractions.
Instructions
Write down addition of fractions. Then, find their least common multiple. To do this, perform the following sequence of actions: 1. Imagine each of the denominators in prime numbers (a prime number, a number that is divisible only by 1 and itself without a remainder, for example 2, 3, 5, 7, etc.).2. Group all the simple ones that are written out, indicating their degrees. 3. Choose the largest powers of each of these prime factors that appear in these numbers. 4. Multiply the written powers.
For example, the common denominator for fractions with denominators 15, 24 and 36 will be a number that can be calculated as follows: 15=3 5; 24=2^3 3;36=2^3 3^2. Write the greatest powers of all prime divisors of these numbers: 2^3 3^2 5=360.
Divide the common denominator by each and the denominators of the fractions being added. Multiply their numerators by the resulting number. Under the common line of the fraction, write the least common dividend, which is also the lowest common denominator. In the numerator, add the numbers that result from multiplying each numerator by the quotient of the least common factor divided by the denominator of the fraction. The sum of all numerators and divided by the lowest common denominator will be the desired number.
For example, for 4/15, 7/24 and 11/36 do this. Find the lowest common denominator, which is 360. Then divide 360/15=24, 360/24=15, 360/36=10. Multiply the number 4, which is the numerator of the first fraction, by 24 (4 24=96), the number 7 by 15 (7 15=105), the number 11 by 10 (11 10=110). Then add these numbers (96+105+110=301). We get the result 4/15+7/24+11/36=301/360.
Sources:
- how to find the smallest number
Integers are a variety of mathematical numbers that have many applications in everyday life. Non-negative integers are used when indicating the number of any objects, negative numbers - in messages about weather forecasts, etc. GCD and LCM are natural characteristics of integers associated with division operations.
Instructions
GCD is easy to calculate using the Euclidean algorithm or the binary method. According to the Euclid algorithm for determining the gcd of numbers a and b, one of which is not zero, there is a sequence of numbers r_1 > r_2 > r_3 > ... > r_n, in which r_1 is equal to the remainder of dividing the first number by the second. And the other members of the sequence are equal to the remainders from dividing the previous member by the previous one, and the penultimate element is divided by the last one without a remainder.
Mathematically, the sequence can be represented as:
a = b*k_0 + r_1
b = r_1*k_1 + r_2
r_1 = r_2*k_2 + r_3
…
r_(n - 1) = r_n*k_n,
where k_i is an integer factor.
GCD (a, b) = r_n.
Example.
Find GCD (36, 120). According to the Euclidean algorithm, subtract from 120 a number that is a multiple of 36, in this case it is 120 – 36*3 = 12. Now subtract a number that is a multiple of 12 from 120, you get 120 – 12*10 = 0. Therefore, GCD (36, 120) = 12.
The binary algorithm for finding GCD is based on shift theory. According to this method, the gcd of two numbers has the following properties:
GCD (a, b) = 2*GCD (a/2, b/2) for even a and b
GCD (a, b) = GCD (a/2, b) for even a and odd b (the opposite is true for GCD (a, b) = GCD (a, b/2))
GCD (a, b) = GCD ((a - b)/2, b) for odd a > b
GCD (a, b) = GCD ((b - a)/2, a) for odd b > a
Thus, gcd (36, 120) = 2*gcd (18, 60) = 4*gcd (9, 30) = 4* gcd (9, 15) = 4*gcd ((15 - 9)/2=3 , 9) = 4*3 = 12.
The least common multiple (LCM) of two integers is the smallest integer that is divisible by both original numbers without leaving a remainder.
The LCM can be calculated using GCD: LCM (a, b) = |a*b|/GCD (a, b).
The second way to calculate the LCM is the canonical factorization of numbers into prime factors:
a = r_1^k_1*…*r_n^k_n
b = r_1^m_1*…*r_n^m_n,
where r_i are prime numbers, and k_i and m_i are integers ≥ 0.
LCM is represented in the form of the same prime factors, where the maximum of two numbers is taken as powers.
Example.
Find the LCM (16, 20):
16 = 2^4*3^0*5^0
20 = 2^2*3^0*5^1
LCM (16, 20) = 2^4*3^0*5^1 = 16*5 = 80.
LCM - least common multiple. A number that will divide all given numbers without a remainder.
For example, if the given numbers are 2, 3, 5, then LCM=2*3*5=30
And if the given numbers are 2,4,8, then LCM =8
what is GCD?
GCD is the greatest common divisor. A number that can be used to divide each of the given numbers without leaving a remainder.
It is logical that if the given numbers are prime, then the gcd is equal to one.
And if the given numbers are 2, 4, 8, then GCD is equal to 2.
We will not describe it in general terms, but will simply show the solution with an example.
Given two numbers 126 and 44. Find GCD.
Then if we are given two numbers of the form
Then GCD is calculated as
where min is the minimum value of all powers of the number pn
and NOC as
where max is the maximum value of all powers of the number pn
Looking at the above formulas, you can easily prove that the gcd of two or more numbers will be equal to one, when among at least one pair of given values there are relatively prime numbers.
Therefore, it is easy to answer the question of what the gcd of such numbers as 3, 25412, 3251, 7841, 25654, 7 is equal to without calculating anything.
numbers 3 and 7 are coprime, and therefore gcd = 1
Let's look at an example.
Given three numbers 24654, 25473 and 954
Each number is decomposed into the following factors
Or, if we write it in an alternative form
That is, the gcd of these three numbers is equal to three
Well, we can calculate the LCM in a similar way, and it is equal to
Our bot will help you calculate the GCD and LCM of any integers, two, three or ten.
The material presented below is a logical continuation of the theory from the article entitled LCM - least common multiple, definition, examples, connection between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and we will pay special attention to solving examples. First, we will show how the LCM of two numbers is calculated using the GCD of these numbers. Next, we'll look at finding the least common multiple by factoring numbers into prime factors. After this, we will focus on finding the LCM of three or more numbers, and also pay attention to calculating the LCM of negative numbers.
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Calculating Least Common Multiple (LCM) via GCD
One way to find the least common multiple is based on the relationship between LCM and GCD. The existing connection between LCM and GCD allows us to calculate the least common multiple of two positive integers through a known greatest common divisor. The corresponding formula is LCM(a, b)=a b:GCD(a, b) . Let's look at examples of finding the LCM using the given formula.
Example.
Find the least common multiple of two numbers 126 and 70.
Solution.
In this example a=126 , b=70 . Let us use the connection between LCM and GCD, expressed by the formula LCM(a, b)=a b:GCD(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 using the written formula.
Let's find GCD(126, 70) using the Euclidean algorithm: 126=70·1+56, 70=56·1+14, 56=14·4, therefore, GCD(126, 70)=14.
Now we find the required least common multiple: GCD(126, 70)=126·70:GCD(126, 70)= 126·70:14=630.
Answer:
LCM(126, 70)=630 .
Example.
What is LCM(68, 34) equal to?
Solution.
Because 68 is divisible by 34, then GCD(68, 34)=34. Now we calculate the least common multiple: GCD(68, 34)=68·34:GCD(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 you compose a product from all the prime factors of given numbers, and then exclude from this product all the common prime factors present in the decompositions of the given numbers, then the resulting product will be equal to the least common multiple of the given numbers.
The stated rule for finding the LCM follows from the equality LCM(a, b)=a b:GCD(a, b). Indeed, the product of numbers a and b is equal to the product of all factors involved in the expansion of numbers a and b. In turn, GCD(a, b) is equal to the product of all prime factors simultaneously present in the expansions of numbers a and b (as described in the section on finding GCD using the expansion of numbers into prime factors).
Let's give an example. Let us know that 75=3·5·5 and 210=2·3·5·7. Let's compose the product from all the factors of these expansions: 2·3·3·5·5·5·7 . Now from this product we exclude all the factors present in both the expansion of the number 75 and the expansion of the number 210 (these 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, NOC(75, 210)= 2·3·5·5·7=1,050.
Example.
Factor the numbers 441 and 700 into prime factors and find the least common multiple of these numbers.
Solution.
Let's factor 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 create a product from all the factors involved in the expansion of these numbers: 2·2·3·3·5·5·7·7·7. Let us exclude from this product all factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2·2·3·3·5·5·7·7. Thus, LCM(441, 700)=2·2·3·3·5·5·7·7=44 100.
Answer:
NOC(441, 700)= 44 100 .
The rule for finding the LCM using factorization of numbers into prime factors can be formulated a little differently. If the missing factors from the expansion of number b are added 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 the same numbers 75 and 210, their decompositions into prime factors are as follows: 75=3·5·5 and 210=2·3·5·7. To the factors 3, 5 and 5 from the expansion of the number 75 we add the missing factors 2 and 7 from the expansion of the number 210, we obtain the product 2·3·5·5·7, the value of which is equal to LCM(75, 210).
Example.
Find the least common multiple of 84 and 648.
Solution.
We first obtain the decompositions of the numbers 84 and 648 into prime factors. They look like 84=2·2·3·7 and 648=2·2·2·3·3·3·3. To the factors 2, 2, 3 and 7 from the expansion of the number 84 we add the missing factors 2, 3, 3 and 3 from the expansion of the number 648, we obtain the product 2 2 2 3 3 3 3 7, which is equal to 4 536 . Thus, the desired least common multiple of 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 sequentially finding the LCM of two numbers. Let us recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.
Theorem.
Let positive integer numbers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found by sequentially calculating m 2 = LCM(a 1 , a 2) , m 3 = LCM(m 2 , a 3) , … , m k = LCM(m k−1 , a k) .
Let's consider the application of this theorem using the example of finding the least common multiple of four numbers.
Example.
Find the LCM of 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 = LOC(a 1 , a 2) = LOC(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 , from where GCD(140, 9)=140 9:GCD(140, 9)= 140·9:1=1,260. That is, m 2 =1 260.
Now we find m 3 = LOC (m 2 , a 3) = LOC (1 260, 54). Let's calculate it through GCD(1 260, 54), which we also determine using the Euclidean algorithm: 1 260=54·23+18, 54=18·3. Then gcd(1,260, 54)=18, from which gcd(1,260, 54)= 1,260·54:gcd(1,260, 54)= 1,260·54:18=3,780. That is, m 3 =3 780.
All that remains is to find m 4 = LOC(m 3, a 4) = LOC(3 780, 250). To do this, we find GCD(3,780, 250) using the Euclidean algorithm: 3,780=250·15+30, 250=30·8+10, 30=10·3. Therefore, GCM(3,780, 250)=10, whence GCM(3,780, 250)= 3 780 250: GCD(3 780, 250)= 3,780·250:10=94,500. That is, m 4 =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, it is convenient to find the least common multiple of three or more numbers using prime factorizations of the given numbers. In this case, you should adhere to the following rule. 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 factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the resulting factors, and so on.
Let's look at an example of finding the least common multiple using prime factorization.
Example.
Find the least common multiple of the five numbers 84, 6, 48, 7, 143.
Solution.
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 decomposition of the number 6 does not contain missing factors, since both 2 and 3 are already present in the decomposition of the first number 84. Next, 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 will be no need to add multipliers 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.
Let's look at three ways to find the least common multiple.
Finding by factorization
The first method is to find the least common multiple by factoring the given numbers into prime factors.
Let's say we need to find the LCM of the numbers: 99, 30 and 28. To do this, let's factor each of these numbers into prime factors:
For the desired number to be divisible by 99, 30 and 28, it is necessary and sufficient that it includes all the prime factors of these divisors. To do this, we need to take all the prime factors of these numbers to the greatest possible power and multiply them together:
2 2 3 2 5 7 11 = 13,860
Thus, LCM (99, 30, 28) = 13,860. No other number less than 13,860 is divisible by 99, 30, or 28.
To find the least common multiple of given numbers, you factor them into their prime factors, then take each prime factor with the largest exponent it appears in, and multiply those factors together.
Since relatively prime numbers do not have common prime factors, their least common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are relatively prime. That's why
LCM (20, 49, 33) = 20 49 33 = 32,340.
The same must be done when finding the least common multiple of various prime numbers. For example, LCM (3, 7, 11) = 3 7 11 = 231.
Finding by selection
The second method is to find the least common multiple by selection.
Example 1. When the largest of given numbers is divided by another given number, then the LCM of these numbers is equal to the largest of them. For example, given four numbers: 60, 30, 10 and 6. Each of them is divisible by 60, therefore:
LCM(60, 30, 10, 6) = 60
In other cases, to find the least common multiple, the following procedure is used:
- Determine the largest number from the given numbers.
- Next, we find the numbers that are multiples of the largest number by multiplying it by natural numbers in increasing order and checking whether the resulting product is divisible by the remaining given numbers.
Example 2. Given three numbers 24, 3 and 18. We determine the largest of them - this is the number 24. Next, we find the numbers that are multiples of 24, checking whether each of them is divisible by 18 and 3:
24 · 1 = 24 - divisible by 3, but not divisible by 18.
24 · 2 = 48 - divisible by 3, but not divisible by 18.
24 · 3 = 72 - divisible by 3 and 18.
Thus, LCM (24, 3, 18) = 72.
Finding by sequentially finding the LCM
The third method is to find the least common multiple by sequentially finding the LCM.
The LCM of two given numbers is equal to the product of these numbers divided by their greatest common divisor.
Example 1. Find the LCM of two given numbers: 12 and 8. Determine their greatest common divisor: GCD (12, 8) = 4. Multiply these numbers:
We divide the product by their gcd:
Thus, LCM (12, 8) = 24.
To find the LCM of three or more numbers, use the following procedure:
- First, find the LCM of any two of these numbers.
- Then, LCM of the found least common multiple and the third given number.
- Then, the LCM of the resulting least common multiple and the fourth number, etc.
- Thus, the search for LCM continues as long as there are numbers.
Example 2. Let's find the LCM of three given numbers: 12, 8 and 9. We already found the LCM of the numbers 12 and 8 in the previous example (this is the number 24). It remains to find the least common multiple of the number 24 and the third given number - 9. Determine their greatest common divisor: GCD (24, 9) = 3. Multiply the LCM with the number 9:
We divide the product by their gcd:
Thus, LCM (12, 8, 9) = 72.
Let's continue the conversation about the least common multiple, which we started in the section “LCM - least common multiple, definition, examples.” In this topic, we will look at ways to find the LCM for three or more numbers, and we will look at the question of how to find the LCM of a negative number.
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Calculating Least Common Multiple (LCM) via GCD
We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to determine the LCM through GCD. First, let's figure out how to do this for positive numbers.
Definition 1
You can find the least common multiple through the greatest common divisor using the formula LCM (a, b) = a · b: GCD (a, b).
Example 1
You need to find the LCM of the numbers 126 and 70.
Solution
Let's take a = 126, b = 70. Let's substitute the values into the formula for calculating the least common multiple through the greatest common divisor LCM (a, b) = a · b: GCD (a, b) .
Finds the gcd of numbers 70 and 126. For this we need the Euclidean algorithm: 126 = 70 1 + 56, 70 = 56 1 + 14, 56 = 14 4, therefore GCD (126 , 70) = 14 .
Let's calculate the LCM: LCD (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.
Answer: LCM(126, 70) = 630.
Example 2
Find the number 68 and 34.
Solution
GCD in this case is not difficult to find, since 68 is divisible by 34. Let's calculate the least common multiple using the formula: LCM (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.
Answer: LCM(68, 34) = 68.
In this example, we used the rule for finding the least common multiple of positive integers a and b: if the first number is divisible by the second, the LCM of those numbers will be equal to the first number.
Finding the LCM by factoring numbers into prime factors
Now let's look at the method of finding the LCM, which is based on factoring numbers into prime factors.
Definition 2
To find the least common multiple, we need to perform a number of simple steps:
- we compose the product of all prime factors of the numbers for which we need to find the LCM;
- we exclude all prime factors from their resulting products;
- the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.
This method of finding the least common multiple is based on the equality LCM (a, b) = a · b: GCD (a, b). If you look at the formula, it will become clear: the product of the numbers a and b is equal to the product of all the factors that participate in the decomposition of these two numbers. In this case, the gcd of two numbers is equal to the product of all prime factors that are simultaneously present in the factorizations of these two numbers.
Example 3
We have two numbers 75 and 210. We can factor them as follows: 75 = 3 5 5 And 210 = 2 3 5 7. If you compose the product of all the factors of the two original numbers, you get: 2 3 3 5 5 5 7.
If we exclude the factors common to both numbers 3 and 5, we get a product of the following form: 2 3 5 5 7 = 1050. This product will be our LCM for the numbers 75 and 210.
Example 4
Find the LCM of numbers 441 And 700 , factoring both numbers into prime factors.
Solution
Let's find all the prime factors of the numbers given in the condition:
441 147 49 7 1 3 3 7 7
700 350 175 35 7 1 2 2 5 5 7
We get two chains of numbers: 441 = 3 3 7 7 and 700 = 2 2 5 5 7.
The product of all factors that participated in the decomposition of these numbers will have the form: 2 2 3 3 5 5 7 7 7. Let's find common factors. This is the number 7. Let's exclude it from the total product: 2 2 3 3 5 5 7 7. It turns out that NOC (441, 700) = 2 2 3 3 5 5 7 7 = 44 100.
Answer: LOC(441, 700) = 44,100.
Let us give another formulation of the method for finding the LCM by decomposing numbers into prime factors.
Definition 3
Previously, we excluded from the total number of factors common to both numbers. Now we will do it differently:
- Let's factor both numbers into prime factors:
- add to the product of the prime factors of the first number the missing factors of the second number;
- we obtain the product, which will be the desired LCM of two numbers.
Example 5
Let's return to the numbers 75 and 210, for which we already looked for the LCM in one of the previous examples. Let's break them down into simple factors: 75 = 3 5 5 And 210 = 2 3 5 7. To the product of factors 3, 5 and 5 numbers 75 add the missing factors 2 And 7 numbers 210. We get: 2 · 3 · 5 · 5 · 7 . This is the LCM of the numbers 75 and 210.
Example 6
It is necessary to calculate the LCM of the numbers 84 and 648.
Solution
Let's factor the numbers from the condition into simple factors: 84 = 2 2 3 7 And 648 = 2 2 2 3 3 3 3. Let's add to the product the factors 2, 2, 3 and 7
numbers 84 missing factors 2, 3, 3 and
3
numbers 648. We get the product 2 2 2 3 3 3 3 7 = 4536. This is the least common multiple of 84 and 648.
Answer: LCM(84, 648) = 4,536.
Finding the LCM of three or more numbers
Regardless of how many numbers we are dealing with, the algorithm of our actions will always be the same: we will sequentially find the LCM of two numbers. There is a theorem for this case.
Theorem 1
Let's assume we have integers a 1 , a 2 , … , a k. NOC m k these numbers are found by sequentially calculating m 2 = LCM (a 1, a 2), m 3 = LCM (m 2, a 3), ..., m k = LCM (m k − 1, a k).
Now let's look at how the theorem can be applied to solve specific problems.
Example 7
You need to calculate the least common multiple of four numbers 140, 9, 54 and 250 .
Solution
Let us introduce the notation: a 1 = 140, a 2 = 9, a 3 = 54, a 4 = 250.
Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140, 9). Let's apply the Euclidean algorithm to calculate the GCD of the numbers 140 and 9: 140 = 9 15 + 5, 9 = 5 1 + 4, 5 = 4 1 + 1, 4 = 1 4. We get: GCD (140, 9) = 1, GCD (140, 9) = 140 9: GCD (140, 9) = 140 9: 1 = 1,260. Therefore, m 2 = 1,260.
Now let’s calculate using the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260, 54). During the calculations we obtain m 3 = 3 780.
We just have to calculate m 4 = LCM (m 3 , a 4) = LCM (3 780, 250). We follow the same algorithm. We get m 4 = 94 500.
The LCM of the four numbers from the example condition is 94500.
Answer: NOC (140, 9, 54, 250) = 94,500.
As you can see, the calculations are simple, but quite labor-intensive. To save time, you can go another way.
Definition 4
We offer you the following algorithm of actions:
- we decompose all numbers into prime factors;
- to the product of the factors of the first number we add the missing factors from the product of the second number;
- to the product obtained at the previous stage we add the missing factors of the third number, etc.;
- the resulting product will be the least common multiple of all numbers from the condition.
Example 8
You need to find the LCM of five numbers 84, 6, 48, 7, 143.
Solution
Let's factor all five numbers into prime factors: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 3, 7, 143 = 11 13. Prime numbers, which is the number 7, cannot be factored into prime factors. Such numbers coincide with their decomposition into prime factors.
Now let's take the product of the prime factors 2, 2, 3 and 7 of the number 84 and add to them the missing factors of the second number. We decomposed the number 6 into 2 and 3. These factors are already in the product of the first number. Therefore, we omit them.
We continue to add the missing multipliers. Let's move on to the number 48, from the product of whose prime factors we take 2 and 2. Then we add the prime factor of 7 from the fourth number and the factors of 11 and 13 of the fifth. We get: 2 2 2 2 3 7 11 13 = 48,048. This is the least common multiple of the original five numbers.
Answer: LCM(84, 6, 48, 7, 143) = 48,048.
Finding the least common multiple of negative numbers
In order to find the least common multiple of negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations must be carried out using the above algorithms.
Example 9
LCM (54, − 34) = LCM (54, 34) and LCM (− 622, − 46, − 54, − 888) = LCM (622, 46, 54, 888).
Such actions are permissible due to the fact that if we accept that a And − a– opposite numbers,
then the set of multiples of a number a matches the set of multiples of a number − a.
Example 10
It is necessary to calculate the LCM of negative numbers − 145 And − 45 .
Solution
Let's replace the numbers − 145 And − 45 to their opposite numbers 145 And 45 . Now, using the algorithm, we calculate the LCM (145, 45) = 145 · 45: GCD (145, 45) = 145 · 45: 5 = 1,305, having previously determined the GCD using the Euclidean algorithm.
We get that the LCM of the numbers is − 145 and − 45 equals 1 305 .
Answer: LCM (− 145, − 45) = 1,305.
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