Least common multiple of a number 2. How to find the least common multiple, but for two or more numbers
How to find the least common multiple?
How to find the NOC
Here is a video that will show you two ways to find the least common multiple (LCM). By practicing using the first of the proposed methods, you can better understand what the least common multiple is.
- We represent each number as the product of its prime factors:
- We write down the powers of all prime factors:
- We select all prime divisors (multipliers) with the largest degrees, multiply them and find the LCM:
- The first step is to decompose these numbers into prime factors.
- We write out the factors that are included in the expansion of the number 30. This is 2 x 3 x 5.
- Now you need to multiply them by the missing factor, which we have when decomposing 42, and this is 7. We get 2 x 3 x 5 x 7.
- We find what is equal to 2 x 3 x 5 x 7 and get 210.
- We decompose both numbers into prime factors: 8=2*2*2 and 12=3*2*2
- We reduce the same multipliers for one of the numbers. In our case, 2 * 2 match, we reduce them for the number 12, then 12 will have one factor: 3.
- Find the product of all remaining factors: 2*2*2*3=24
It is necessary to find each factor of each of the two numbers for which we find the least common multiple, and then multiply the factors that coincided with the first and second numbers by each other. The result of the product will be the desired multiple.
For example, we have the numbers 3 and 5 and we need to find the LCM (least common multiple). Us must be multiplied and three and five for all numbers starting from 1 2 3 ... and so on until we see the same number here and there.
We multiply the three and get: 3, 6, 9, 12, 15
Multiply five and get: 5, 10, 15
The prime factorization method is the most classic for finding the least common multiple (LCM) of multiple numbers. This method is clearly and simply demonstrated in the following video:
Add, multiply, divide, reduce to a common denominator and others arithmetic operations a very exciting activity, examples that occupy a whole sheet are especially admired.
So find the common multiple for two numbers, which will be the smallest number by which two numbers are divisible. I want to note that it is not necessary to resort to formulas in the future to find what you are looking for, if you can count in your mind (and this can be trained), then the numbers themselves pop up in your head and then the fractions click like nuts.
To begin with, we will learn that we can multiply two numbers against each other, and then reduce this figure and divide alternately by these two numbers, so we will find the smallest multiple.
For example, two numbers 15 and 6. We multiply and get 90. This is clearly more number. Moreover, 15 is divisible by 3 and 6 is divisible by 3, which means we also divide 90 by 3. We get 30. We try to divide 30 by 15 is 2. And 30 divides 6 is 5. Since 2 is the limit, it turns out that the smallest multiple for the numbers 15 and 6 will be 30.
With more numbers it will be a little more difficult. but if you know which numbers give a zero remainder when divided or multiplied, then, in principle, there are no big difficulties.
Here's another way to find the least common multiple. Let's take a look at an illustrative example.
It is necessary to find the LCM of three numbers at once: 16, 20 and 28.
16 = 224 = 2^24^1
20 = 225 = 2^25^1
28 = 227 = 2^27^1
LCM = 2^24^15^17^1 = 4457 = 560.
LCM(16, 20, 28) = 560.
Thus, as a result of the calculation, the number 560 was obtained. It is the least common multiple, that is, it is divisible by each of the three numbers without a remainder.
The least common multiple is the number that can be divided by several given numbers without a remainder. In order to calculate such a figure, you need to take each number and decompose it into simple factors. Those numbers that match are removed. Leaves everyone one at a time, multiply them among themselves in turn and get the desired - the least common multiple.
NOC, or least common multiple, is the smallest natural number two or more numbers that is divisible by each of the given numbers without a remainder.
Here is an example of how to find the least common multiple of 30 and 42.
For 30, it's 2 x 3 x 5.
For 42, this is 2 x 3 x 7. Since 2 and 3 are in the expansion of the number 30, we cross them out.
As a result, we get that the LCM of the numbers 30 and 42 is 210.
To find the least common multiple, you need to follow a few simple steps in sequence. Consider this using the example of two numbers: 8 and 12
Checking, we make sure that 24 is divisible by both 8 and 12, and this is the smallest natural number that is divisible by each of these numbers. Here we are find the least common multiple.
I'll try to explain using the example of the numbers 6 and 8. The least common multiple is the number that can be divided by these numbers (in our case, 6 and 8) and there will be no remainder.
So, we begin to multiply first 6 by 1, 2, 3, etc. and 8 by 1, 2, 3, etc.
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
Finding a common denominator when adding and subtracting fractions different denominators need to know and be able to calculate least common multiple(NOC).
A multiple of the number "a" is a number that is itself divisible by the number "a" without a remainder.
Numbers that are multiples of 8 (that is, these numbers will be divided by 8 without a remainder): these are the numbers 16, 24, 32 ...
Multiples of 9: 18, 27, 36, 45…
There are infinitely many multiples of a given number a, in contrast to the divisors of the same number. Divisors - a finite number.
A common multiple of two natural numbers is a number that is evenly divisible by both of these numbers..
Least common multiple(LCM) of two or more natural numbers is the smallest natural number that is itself divisible by each of these numbers.
How to find the NOC
LCM can be found and written in two ways.
The first way to find the LCM
This method is usually used for small numbers.
- We write the multiples for each of the numbers in a line until there is a multiple that is the same for both numbers.
- A multiple of the number "a" is denoted by a capital letter "K".
Example. Find LCM 6 and 8.
The second way to find the LCM
This method is convenient to use to find the LCM for three or more numbers.
The number of identical factors in the expansions of numbers can be different.
LCM (24, 60) = 2 2 3 5 2
Answer: LCM (24, 60) = 120
You can also formalize finding the least common multiple (LCM) as follows. Let's find the LCM (12, 16, 24) .
24 = 2 2 2 3
As we can see from the expansion of numbers, all factors of 12 are included in the expansion of 24 (the largest of the numbers), so we add only one 2 from the expansion of the number 16 to the LCM.
LCM (12, 16, 24) = 2 2 2 3 2 = 48
Answer: LCM (12, 16, 24) = 48
Special cases of finding NOCs
For example, LCM(60, 15) = 60
Since coprime numbers have no common prime divisors, their least common multiple is equal to the product of these numbers.
On our site, you can also use a special calculator to find the least common multiple online to check your calculations.
If a natural number is only divisible by 1 and itself, then it is called prime.
Any natural number is always divisible by 1 and itself.
The number 2 is the smallest prime number. This is the only even prime number, the rest of the prime numbers are odd.
There are many prime numbers, and the first among them is the number 2. However, there is no last prime number. In the "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 largest number, by which both numbers "a" and "b" are divisible without a remainder.
Briefly, the greatest common divisor of numbers "a" and "b" is written as follows:
Example: gcd (12; 36) = 12 .
The divisors of numbers in the solution record are denoted by a capital letter "D".
The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.
Coprime numbers are natural numbers that have only one common divisor - the number 1. Their GCD is 1.
How to find the greatest common divisor
To find the gcd of two or more natural numbers you need:
Calculations are conveniently written using a vertical bar. To the left of the line, first write down the dividend, to the right - the divisor. Further in the left column we write down the values of private.
Let's explain right away with an example. Let's factorize the numbers 28 and 64 into prime factors.
- Underline the same prime factors in both numbers.
28 = 2 2 7
64 = 2 2 2 2 2 2
We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4
Answer: GCD (28; 64) = 4
You can arrange the location of the GCD in two ways: in a column (as was done above) or “in a line”.
The first way to write GCD
Find GCD 48 and 36.
GCD (48; 36) = 2 2 3 = 12
The second way to write GCD
Now let's write the GCD search solution in a line. Find GCD 10 and 15.
On our information site, you can also find the greatest common divisor online using the helper program to check your calculations.
Finding the least common multiple, methods, examples of finding the LCM.
The material presented below is a logical continuation of the theory from the article under the heading LCM - Least Common Multiple, definition, examples, relationship between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and Special attention Let's take a look at the examples. Let us first show how the LCM of two numbers is calculated in terms of the GCD of these numbers. Next, consider finding the least common multiple by factoring numbers into prime factors. After that, we will focus on finding the LCM of three and more numbers, and also pay attention to the calculation of the LCM of negative numbers.
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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: GCD(a, b). Consider examples of finding the LCM according to the above formula.
Find the least common multiple of the two numbers 126 and 70 .
In this example a=126 , b=70 . Let's use the link of LCM with GCD, which is expressed by the formula LCM(a, b)=a b: GCM(a, b) . That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers according to the written formula.
Find gcd(126, 70) using Euclid's algorithm: 126=70 1+56 , 70=56 1+14 , 56=14 4 , hence gcd(126, 70)=14 .
Now we find the required least common multiple: LCM(126, 70)=126 70:GCD(126, 70)= 126 70:14=630 .
What is LCM(68, 34) ?
Since 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34:GCD(68, 34)= 68 34:34=68 .
Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b , then the least common multiple of these numbers is a .
Finding the LCM by Factoring Numbers into Prime Factors
Another way to find the least common multiple is based on factoring numbers into prime factors. If we make a product of all prime factors of these numbers, after which we exclude from this product all common prime factors that are present in the expansions of these numbers, then the resulting product will be equal to the least common multiple of these numbers.
The announced rule for finding the LCM follows from the equality LCM(a, b)=a b: GCM(a, b) . Indeed, the product of the numbers a and b is equal to the product of all the factors involved in the expansions of the numbers a and b. In turn, gcd(a, b) is equal to the product of all prime factors that are simultaneously present in the expansions of the numbers a and b (which is described in the section on finding the gcd using the decomposition of numbers into prime factors).
Let's take an example. Let we know that 75=3 5 5 and 210=2 3 5 7 . Compose the product of all factors of these expansions: 2 3 3 5 5 5 7 . Now we exclude from this product all the factors that are present both in the expansion of the number 75 and in the expansion of the number 210 (such factors are 3 and 5), then the product will take the form 2 3 5 5 7 . The value of this product is equal to the least common multiple of 75 and 210 , that is, LCM(75, 210)= 2 3 5 5 7=1 050 .
After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.
Let's decompose the numbers 441 and 700 into prime factors: 
We get 441=3 3 7 7 and 700=2 2 5 5 7 .
Now let's make a product of all the factors involved in the expansions of these numbers: 2 2 3 3 5 5 7 7 7 . Let us exclude from this product all 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 expansion of the number 75, we add the missing factors 2 and 7 from the expansion of the number 210, we get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .
Find the least common multiple of 84 and 648.
We first obtain the decomposition of the numbers 84 and 648 into prime factors. They look like 84=2 2 3 7 and 648=2 2 2 3 3 3 3 . To the factors 2 , 2 , 3 and 7 from the expansion of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the expansion of the number 648 , we get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 84 and 648 is 4,536.
Finding the LCM of three or more numbers
The least common multiple of three or more numbers can be found by successively finding the LCM of two numbers. Recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.
Let positive integers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k =LCM(m k−1 , a k) .
Consider the application of this theorem on the example of finding the least common multiple of four numbers.
Find the LCM of the four numbers 140 , 9 , 54 and 250 .
First we find m 2 = LCM (a 1 , a 2) = LCM (140, 9) . To do this, using the Euclidean algorithm, we determine gcd(140, 9) , we have 140=9 15+5 , 9=5 1+4 , 5=4 1+1 , 4=1 4 , therefore, gcd(140, 9)=1 , whence LCM(140, 9)=140 9: GCD(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .
Now we find m 3 = LCM (m 2 , a 3) = LCM (1 260, 54) . Let's calculate it through gcd(1 260, 54) , which is also determined by the Euclid algorithm: 1 260=54 23+18 , 54=18 3 . Then gcd(1 260, 54)=18 , whence LCM(1 260, 54)= 1 260 54:gcd(1 260, 54)= 1 260 54:18=3 780 . That is, m 3 \u003d 3 780.
It remains to find m 4 = LCM (m 3 , a 4) = LCM (3 780, 250) . To do this, we find GCD(3 780, 250) using the Euclid algorithm: 3 780=250 15+30 , 250=30 8+10 , 30=10 3 . Therefore, gcd(3 780, 250)=10 , hence LCM(3 780, 250)= 3 780 250:gcd(3 780, 250)= 3 780 250:10=94 500 . That is, m 4 \u003d 94 500.
So the least common multiple of the original four numbers is 94,500.
LCM(140, 9, 54, 250)=94500 .
In many cases, the least common multiple of three or more numbers is conveniently found using prime factorizations of given numbers. At the same time, one should adhere to next 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 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. essence this method is to factor both numbers into prime factors and multiply the common ones.
Example 1. Find GCD of numbers 24 and 18
First, let's factor both numbers into prime factors:
Now we multiply their common factors. In order not to get confused, the common factors can be underlined.
We look at the decomposition of the number 24. Its first factor is 2. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We underline both twos:
Again we look at the decomposition of the number 24. Its second factor is also 2. We are looking for the same factor in the decomposition of the number 18 and see that it is not there for the second time. Then we don't highlight anything.
The next two in the expansion of the number 24 is also missing in the expansion of the number 18.
We pass to the last factor in the decomposition of the number 24. This is the factor 3. We are looking for the same factor in the decomposition of the number 18 and we see that it is also there. We emphasize both threes:
So, the common factors of the numbers 24 and 18 are the factors 2 and 3. To get the GCD, these factors must be multiplied:
So gcd (24 and 18) = 6
The third way to find GCD
Now consider the third way to find the greatest common divisor. The essence of this method lies in the fact that the numbers to be searched for the greatest common divisor are decomposed into prime factors. Then, from the decomposition of the first number, factors that are not included in the decomposition of the second number are deleted. The remaining numbers in the first expansion are multiplied and get GCD.
For example, let's find the GCD for the numbers 28 and 16 in this way. First of all, we decompose these numbers into prime factors:
We got two expansions: and
Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include seven. We will delete it from the first expansion:
Now we multiply the remaining factors and get the GCD:
The number 4 is the greatest common divisor of the numbers 28 and 16. Both of these numbers are divisible by 4 without a remainder:
Example 2 Find GCD of numbers 100 and 40
Factoring out the number 100
Factoring out the number 40
We got two expansions:
Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include one five (there is only one five). We delete it from the first decomposition
Multiply the remaining numbers:
We got the answer 20. So the number 20 is the greatest common divisor of the numbers 100 and 40. These two numbers are divisible by 20 without a remainder:
GCD (100 and 40) = 20.
Example 3 Find the gcd of the numbers 72 and 128
Factoring out the number 72
Factoring out the number 128
2×2×2×2×2×2×2
Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include two triplets (there are none at all). We delete them from the first expansion:
We got the answer 8. So the number 8 is the greatest common divisor of the numbers 72 and 128. These two numbers are divisible by 8 without a remainder:
GCD (72 and 128) = 8
Finding GCD for Multiple Numbers
The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found.
For example, let's find the GCD for the numbers 18, 24 and 36
Factoring the number 18
Factoring the number 24
Factoring the number 36
We got three expansions:
Now we select and underline the common factors in these numbers. Common factors must be included in all three numbers:
We see that the common factors for the numbers 18, 24 and 36 are factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:
We got the answer 6. So the number 6 is the greatest common divisor of the numbers 18, 24 and 36. These three numbers are divisible by 6 without a remainder:
GCD (18, 24 and 36) = 6
Example 2 Find gcd for numbers 12, 24, 36 and 42
Let's factorize each number. Then we find the product of the common factors of these numbers.
Factoring the number 12
Factoring the number 42
We got four expansions:
Now we select and underline the common factors in these numbers. Common factors must be included in all four numbers:
We see that the common factors for the numbers 12, 24, 36, and 42 are the factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:
We got the answer 6. So the number 6 is the greatest common divisor of the numbers 12, 24, 36 and 42. These numbers are divisible by 6 without a remainder:
gcd(12, 24, 36 and 42) = 6
From the previous lesson, we know that if some number is divided by another without a remainder, it is called a multiple of this number.
It turns out that a multiple can be common to several numbers. And now we will be interested in a multiple of two numbers, while it should be as small as possible.
Definition. Least common multiple (LCM) of numbers a and b- a and b a and number b.
Definition contains two variables a and b. Let's substitute any two numbers for these variables. For example, instead of a variable a substitute the number 9, and instead of the variable b let's substitute the number 12. Now let's try to read the definition:
Least common multiple (LCM) of numbers 9 and 12 - This smallest number, which is a multiple 9 and 12 . In other words, it is such a small number that is divisible without a remainder by the number 9 and on the number 12 .
It is clear from the definition that the LCM is the smallest number that is divisible without a remainder by 9 and 12. This LCM is required to be found.
There are two ways to find the least common multiple (LCM). The first way is that you can write down the first multiples of two numbers, and then choose among these multiples such a number that will be common to both numbers and small. Let's apply this method.
First of all, let's find the first multiples for the number 9. To find the multiples for 9, you need to multiply this nine by the numbers from 1 to 9 in turn. The answers you get will be multiples of the number 9. So, let's start. Multiples will be highlighted in red:
Now we find multiples for the number 12. To do this, we multiply 12 by all the numbers 1 to 12 in turn.
Consider the solution of the following problem. The boy's step is 75 cm, and the girl's step is 60 cm. It is necessary to find the smallest distance at which both of them will take an integer number of steps.
Decision. The entire path that the guys will go through must be divisible by 60 and 70 without a remainder, since they must each take an integer number of steps. In other words, the answer must be a multiple of both 75 and 60.
First, we will write out all multiples, for the number 75. We get:
- 75, 150, 225, 300, 375, 450, 525, 600, 675, … .
Now let's write out the numbers that will be a multiple of 60. We get:
- 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, … .
Now we find the numbers that are in both rows.
- Common multiples of numbers will be numbers, 300, 600, etc.
The smallest of them is the number 300. In this case, it will be called the least common multiple of the numbers 75 and 60.
Returning to the condition of the problem, the smallest distance at which the guys take an integer number of steps will be 300 cm. The boy will go this way in 4 steps, and the girl will need to take 5 steps.
Finding the Least Common Multiple
- The least common multiple of two natural numbers a and b is the smallest natural number that is a multiple of both a and b.
In order to find the least common multiple of two numbers, it is not necessary to write down all the multiples for these numbers in a row.
You can use the following method.
How to find the least common multiple
First, you need to decompose these numbers into prime factors.
- 60 = 2*2*3*5,
- 75=3*5*5.
Now let's write down all the factors that are in the expansion of the first number (2,2,3,5) and add to it all the missing factors from the expansion of the second number (5).
As a result, we get a series of prime numbers: 2,2,3,5,5. The product of these numbers will be the least common factor for these numbers. 2*2*3*5*5 = 300.
General scheme for finding the least common multiple
- 1. Decompose numbers into prime factors.
- 2. Write down the prime factors that are part of one of them.
- 3. Add to these factors all those that are in the decomposition of the rest, but not in the selected one.
- 4. Find the product of all the factors written out.
This method is universal. It can be used to find the least common multiple of any number of natural numbers.
The online calculator allows you to quickly find the greatest common divisor and least common multiple of two or any other number of numbers.
Calculator for finding GCD and NOC
Find GCD and NOC
GCD and NOC found: 6433
How to use the calculator
- Enter numbers in the input field
- In case of entering incorrect characters, the input field will be highlighted in red
- press the button "Find GCD and NOC"
How to enter numbers
- Numbers are entered separated by spaces, dots or commas
- The length of the entered numbers is not limited, so finding the gcd and lcm of long numbers will not be difficult
What is NOD and NOK?
Greatest Common Divisor of several numbers is the largest natural integer by which all the original numbers are divisible without a remainder. The greatest common divisor is abbreviated as GCD.
Least common multiple several numbers is the smallest number that is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.
How to check if a number is divisible by another number without a remainder?
To find out if one number is divisible by another without a remainder, you can use some properties of divisibility of numbers. Then, by combining them, one can check the divisibility by some of them and their combinations.
Some signs of divisibility of numbers
1. Sign of divisibility of a number by 2
To determine whether a number is divisible by two (whether it is even), it is enough to look at the last digit of this number: if it is equal to 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if the number 34938 is divisible by 2.
Decision: look at the last digit: 8 means the number is divisible by two.
2. Sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by 3. Thus, to determine whether a number is divisible by 3, you need to calculate the sum of the digits and check if it is divisible by 3. Even if the sum of the digits turned out to be very large, you can repeat the same process again.
Example: determine if the number 34938 is divisible by 3.
Decision: we count the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 3, which means that the number is divisible by three.
3. Sign of divisibility of a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine if the number 34938 is divisible by 5.
Decision: look at the last digit: 8 means the number is NOT divisible by five.
4. Sign of divisibility of a number by 9
This sign is very similar to the sign of divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if the number 34938 is divisible by 9.
Decision: we calculate the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 9, which means that the number is divisible by nine.
How to find GCD and LCM of two numbers
How to find the GCD of two numbers
Most in a simple way calculating the greatest common divisor of two numbers is to find all possible divisors of those numbers and choose the largest of them.
Consider this method using the example of finding GCD(28, 36) :
- We factorize both numbers: 28 = 1 2 2 7 , 36 = 1 2 2 3 3
- We find common factors, that is, those that both numbers have: 1, 2 and 2.
- We calculate the product of these factors: 1 2 2 \u003d 4 - this is the greatest common divisor of the numbers 28 and 36.
How to find the LCM of two numbers
There are two most common ways to find the smallest multiple of two numbers. The first way is that you can write out the first multiples of two numbers, and then choose among them such a number that will be common to both numbers and at the same time the smallest. And the second is to find the GCD of these numbers. Let's just consider it.
To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. Let's find the LCM for the same numbers 28 and 36:
- Find the product of the numbers 28 and 36: 28 36 = 1008
- gcd(28, 36) is already known to be 4
- LCM(28, 36) = 1008 / 4 = 252 .
Finding GCD and LCM for Multiple Numbers
The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. Also, to find the GCD of several numbers, you can use the following relationship: gcd(a, b, c) = gcd(gcd(a, b), c).
A similar relation also applies to the least common multiple of numbers: LCM(a, b, c) = LCM(LCM(a, b), c)
Example: find GCD and LCM for numbers 12, 32 and 36.
- First, let's factorize the numbers: 12 = 1 2 2 3 , 32 = 1 2 2 2 2 2 , 36 = 1 2 2 3 3 .
- Let's find common factors: 1, 2 and 2 .
- Their product will give gcd: 1 2 2 = 4
- Now let's find the LCM: for this we first find the LCM(12, 32): 12 32 / 4 = 96 .
- To find the LCM of all three numbers, you need to find the GCD(96, 36): 96 = 1 2 2 2 2 2 3 , 36 = 1 2 2 3 3 , GCD = 1 2 . 2 3 = 12 .
- LCM(12, 32, 36) = 96 36 / 12 = 288 .
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: GCD(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 .
Decision.
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: 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 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) ?
Decision.
As 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: 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 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.
Decision.
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 . Thus, 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 decomposition of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.
For example, let's take all the same numbers 75 and 210, their expansions into prime factors are as follows: 75=3 5 5 and 210=2 3 5 7 . To the factors 3, 5 and 5 from the expansion of the number 75, we add the missing factors 2 and 7 from the expansion of the number 210, we get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .
Example.
Find the least common multiple of 84 and 648.
Decision.
We first obtain the decomposition of the numbers 84 and 648 into prime factors. They look like 84=2 2 3 7 and 648=2 2 2 3 3 3 3 . To the factors 2 , 2 , 3 and 7 from the expansion of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the expansion of the number 648 , we get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 84 and 648 is 4,536.
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 .
Decision.
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 .
Decision.
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 .
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