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The online calculator allows you to quickly find the greatest common divisor and least common multiple for both two and for any other number of numbers.
Calculator for finding GCD and LCM
Find GCD and LCM
Found GCD and NOC: 5806
How to use the calculator
- Enter numbers in the input field
- If you enter incorrect characters, the input field will be highlighted in red
- click the button "Find GCD and LCM"
How to enter numbers
- Numbers are entered separated by a space, period or comma
- The length of the entered numbers is not limited, so finding the GCD and LCM of long numbers will not be difficult
What are GCD and NOC?
Greatest common divisor multiple numbers is the largest natural integer by which all original numbers are divisible without a remainder. The greatest common factor is abbreviated as Gcd.
Least common multiple multiple 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 that a number is divisible by another number without a remainder?
To find out whether one number is divisible by another without a remainder, you can use some of the divisibility properties of numbers. Then, by combining them, one can check divisibility into some of them and their combinations.
Some signs of divisibility of numbers
1. The criterion for 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 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if 34938 is divisible by 2.
Solution: look at the last digit: 8 - so the number is divisible by two.
2. The sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine if 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 is very large, you can repeat the same process again.
Example: determine if 34938 is divisible by 3.
Solution: 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. The 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 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.
4. The sign of divisibility of a number by 9
This feature is very similar to the divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if 34938 is divisible by 9.
Solution: we count 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
The easiest way to calculate the greatest common divisor of two numbers is to find all possible divisors of those numbers and choose the largest one.
Let us consider this method using the example of finding the GCD (28, 36):
- Factor both numbers: 28 = 1 2 2 7, 36 = 1 2 2 3 3
- We find the common factors, that is, those that both numbers have: 1, 2 and 2.
- We calculate the product of these factors: 1 · 2 · 2 = 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 least 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 consider only it.
To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. 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), as is already known, is equal to 4
- LCM (28, 36) = 1008/4 = 252.
Finding GCD and LCM for several numbers
The greatest common factor can be found for several numbers, not just two. For this, the numbers to be searched for the greatest common factor 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 relationship is valid for 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, factor the numbers: 12 = 1 2 2 3, 32 = 1 2 2 2 2 2 2, 36 = 1 2 2 3 3 3.
- Let's find the common factors: 1, 2 and 2.
- Their product will give GCD: 1 2 2 = 4
- Let us now 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 2 3, 36 = 1 2 2 3 3, GCD = 1 2 2 3 = 12.
- LCM (12, 32, 36) = 96 36/12 = 288.
The topic "Multiples" is studied in the 5th grade of a comprehensive school. Its goal is to improve the written and oral skills of mathematical calculations. In this lesson, new concepts are introduced - "multiples" and "divisors", the technique of finding divisors and multiples of a natural number is being worked out, the ability to find LCM in various ways.
This topic is very important. Knowledge on it can be applied when solving examples with fractions. To do this, you need to find the common denominator by calculating the least common multiple (LCM).
A multiple of A is an integer that is divisible by A without a remainder.
Each natural number has an infinite number of multiples of it. It itself is considered the smallest. The multiple cannot be less than the number itself.
We need to prove that 125 is a multiple of 5. To do this, divide the first number by the second. If 125 is divisible by 5 without a remainder, then the answer is yes.
This method is applicable for small numbers.
There are special cases when calculating the LCM.
1. If you need to find a common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divided without a remainder by the other (20), then this number (80) is the smallest multiple of these two numbers.
LCM (80, 20) = 80.
2. If two do not have a common divisor, then we can say that their LCM is the product of these two numbers.
LCM (6, 7) = 42.
Let's take a look at the last example. 6 and 7 with respect to 42 are divisors. They divide a multiple without a remainder.
In this example, 6 and 7 are paired divisors. Their product is equal to the most multiple of the number (42).
A number is called prime if it is divisible only by itself or by 1 (3: 1 = 3; 3: 3 = 1). The rest are called composite.
In another example, you need to determine if 9 is a divisor of 42.
42: 9 = 4 (remainder 6)
Answer: 9 is not a divisor of 42, because there is a remainder in the answer.
The divisor differs from the multiple in that the divisor is the number by which the natural numbers are divided, and the multiple itself is divisible by this number.
Greatest common divisor of numbers a and b, multiplied by their smallest multiple, will give the product of the numbers themselves a and b.
Namely: GCD (a, b) x LCM (a, b) = a x b.
Common multiples for more complex numbers are found in the following way.
For example, find the LCM for 168, 180, 3024.
We decompose these numbers into prime factors, write them in the form of a product of degrees:
168 = 2³х3¹х7¹
2⁴х3³х5¹х7¹ = 15120
LCM (168, 180, 3024) = 15120.
But many natural numbers are evenly divisible by other natural numbers.
For example:
The number 12 is divided by 1, by 2, by 3, by 4, by 6, by 12;
The number 36 is divisible by 1, 2, 3, 4, 6, 12, 18, 36.
The numbers by which the number is evenly divisible (for 12 these are 1, 2, 3, 4, 6 and 12) are called divisors... Natural number divisor a is a natural number that divides a given number a without a remainder. A natural number that has more than two divisors is called composite .
Note that the numbers 12 and 36 have common factors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. Common divisor of two given numbers a and b- this is the number by which both given numbers are divisible without a remainder a and b.
Common multiple multiple numbers is a number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all j total multiples, there is always the smallest one, in this case it is 90. This number is called the smallestcommon multiple (LCM).
The LCM is always a natural number, which must be greater than the largest of the numbers for which it is determined.
Least Common Multiple (LCM). Properties.
Commutability:
Associativity:
In particular, if and are coprime numbers, then:
Least common multiple of two integers m and n is the divisor of all other common multiples m and n... Moreover, the set of common multiples m, n coincides with the set of multiples for LCM ( m, n).
The asymptotics for can be expressed in terms of some number-theoretic functions.
So, Chebyshev function... And:
This follows from the definition and properties of the Landau function g (n).
What follows from the law of distribution of prime numbers.
Finding the least common multiple (LCM).
LCM ( a, b) can be calculated in several ways:
1. If the greatest common divisor is known, you can use its relationship with the LCM:
2. Let the canonical decomposition of both numbers into prime factors be known:
where p 1, ..., p k- various primes, and d 1, ..., d k and e 1, ..., e k- non-negative integers (they can be zeros if the corresponding prime is absent in the decomposition).
Then LCM ( a,b) is calculated by the formula:
In other words, the LCM decomposition contains all prime factors included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.
Example:
The calculation of the least common multiple of several numbers can be reduced to several consecutive calculations of the LCM of two numbers:
Rule. To find the LCM of a series of numbers, you need:
- decompose numbers into prime factors;
- transfer the largest expansion into the factors of the desired product (the product of the factors of the largest number of the given ones), and then add the factors from the expansion of other numbers that do not occur in the first number or appear in it fewer times;
- the resulting product of prime factors will be the LCM of the given numbers.
Any two or more natural numbers have their LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.
The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.
The prime factors of the largest number 30 were supplemented with a factor of 5 of 25, the resulting product 150 is greater than the largest number 30 and is divided by all given numbers without a remainder. This is the smallest possible product (150, 250, 300 ...), which is a multiple of all given numbers.
The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.
The rule... To calculate the LCM of prime numbers, you need to multiply all these numbers among themselves.
Another option:
To find the least common multiple (LCM) of several numbers, you need:
1) represent each number as a product of its prime factors, for example:
504 = 2 2 2 3 3 7,
2) write down the powers of all prime factors:
504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,
3) write down all the prime divisors (factors) of each of these numbers;
4) choose the highest degree of each of them, found in all expansions of these numbers;
5) multiply these degrees.
Example... Find the LCM of numbers: 168, 180 and 3024.
Solution... 168 = 2 2 2 3 7 = 2 3 3 1 7 1,
180 = 2 2 3 3 5 = 2 2 3 2 5 1,
3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1.
We write out the greatest powers of all prime factors and multiply them:
LCM = 2 4 3 3 5 1 7 1 = 15 120.
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 $ a $ is called a multiple of $ b $.
Let $ a $ and $ b $ be natural numbers. The number $ c $ is called the common divisor for both $ a $ and $ b $.
The set of common divisors for $ a $ and $ b $ is finite, since none of these divisors can be greater than $ a $. This means that among these divisors there is a greatest, 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, you need:
- Find the product of the numbers found in step 2. The resulting number will be the desired greatest common factor.
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 numbers that are included in the decomposition 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 factor.
$ Gcd = 2 \ cdot 11 = $ 22
Example 2
Find the GCD of the monomials $ 63 and $ 81.
We will find according to the presented algorithm. For this:
Decompose numbers into prime factors
$ 63 = 3 \ cdot 3 \ cdot 7 $
$ 81 = 3 \ cdot 3 \ cdot 3 \ cdot 3 $
We choose numbers that are included in the decomposition of these numbers
$ 63 = 3 \ cdot 3 \ cdot 7 $
$ 81 = 3 \ cdot 3 \ cdot 3 \ cdot 3 $
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 the number $ 48 $: $ \ left \ ((\ rm 1,2,3.4.6,8,12,16,24,48) \ right \) $
Now we find the set of divisors of the number $ 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 the given set will be the number $ 12 $. So the greatest common divisor of the numbers $ 48 and $ 60 will be $ 12.
Definition of the LCM
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:
- Factor numbers
- Write down 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 into the first
Example 4
Find the LCM of the numbers $ 99 $ and $ 77 $.
We will find according to the presented algorithm. For this
Factor numbers
$ 99 = 3 \ cdot 3 \ cdot 11 $
Write out the factors included in the first
add to them the factors that are part of the second and do not go into the first
Find the product of the numbers found in Step 2. The resulting number will be the desired least common multiple
$ LCM = 3 \ cdot 3 \ cdot 11 \ cdot 7 = 693 $
Compiling lists of number divisors is often very time consuming. There is a way to find GCD, called Euclid's algorithm.
The statements on which the Euclidean 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 considered numbers until we reach such a pair of numbers 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 $ is a 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 К (a; b) = ab $
Any common divisor of the numbers $ a $ and $ b $ is a divisor of the number $ D (a; b) $
How do I find the least common multiple?
How to find the NOC
Here's a video that shows you two ways to find the least common multiple (LCM). By practicing using the first of these 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 (factors) with the highest powers, multiply them and find the LCM:
- The first step is to factor these numbers into prime factors.
- We write out the factors that are included in the decomposition of the number 30. This is 2 x 3 x 5.
- Now you need to multiply them by the missing factor, which we have in the decomposition of 42, and this is 7. We get 2 x 3 x 5 x 7.
- Find what is 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
- Reduce the same factors for one of the numbers. In our case, 2 * 2 coincide, we will 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
You need to find each factor of each of the two numbers for which we find the smallest common multiple, and then multiply the factors that coincide in the first and second numbers by each other. The result of the product will be the desired multiple.
For example, we have numbers 3 and 5 and we need to find the LCM (least common multiple). US need to multiply and three and five for all numbers starting from 1 2 3 ... and so on until we see the same number both there and there.
We multiply three and get: 3, 6, 9, 12, 15
We multiply the heel and get: 5, 10, 15
The prime factorization method is the most classic for finding the least common multiple (LCM) for multiple numbers. This method is clearly and simply demonstrated in the following video:
Adding, multiplying, dividing, reducing to a common denominator and other arithmetic operations is a very exciting exercise, especially the examples that take up a whole sheet are fascinating.
So find the common multiple of two numbers, which will be the smallest number that divides two numbers. 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, let's learn that you can multiply two numbers by each other, and then reduce this figure and divide it alternately by these two numbers, so we will find the smallest multiple.
For example, two numbers 15 and 6. Multiply and get 90. This is clearly a larger number. Moreover, 15 is divided by 3 and 6 is divided by 3, so 90 is also divided by 3. We get 30. Trying 30 to divide 15 is 2. And 30 divide 6 is 5. Since 2 is the limit, it turns out that the smallest multiple for numbers 15 and 6 would be 30.
Bigger numbers will be a little more difficult. but if you know which numbers give a zero remainder when dividing or multiplying, then, in principle, there are no big difficulties.
Here's another way to find the least common multiple. Let's consider it with 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 smallest common multiple, that is, it is divided by each of the three numbers without a remainder.
The smallest common multiple is a number that can be divided into several suggested numbers without a remainder. In order to calculate such a figure, you need to take each number and decompose it into prime factors. We remove those numbers that match. Leaves all one at a time, multiply them among themselves in turn and get the desired one - the least common multiple.
NOC, or least common multiple, is the smallest natural number of two or more numbers, which is divisible by each of these numbers without a remainder.
Here's an example of how to find the least common multiple of 30 and 42.
For 30, this is 2 x 3 x 5.
For 42 - this is 2 x 3 x 7. Since 2 and 3 are in the decomposition of the number 30, we delete them.
As a result, we get that the LCM of 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 two numbers as an example: 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 found the least common multiple.
I'll try to explain it using the example of the numbers 6 and 8. The smallest common multiple is a 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.
