Least common multiple of numbers 16 and 32. Finding the least common multiple: methods, examples of finding LCM


Largest common divisor and least common multiple are key arithmetic concepts that allow you to operate effortlessly ordinary fractions. LCM and are most often used to find the common denominator of several fractions.

Basic Concepts

The divisor of an integer X is another integer Y by which X is divided without leaving a remainder. For example, the divisor of 4 is 2, and 36 is 4, 6, 9. A multiple of an integer X is a number Y that is divisible by X without a remainder. For example, 3 is a multiple of 15, and 6 is a multiple of 12.

For any pair of numbers we can find their common divisors and multiples. For example, for 6 and 9, the common multiple is 18, and the common divisor is 3. Obviously, pairs can have several divisors and multiples, so the calculations use the largest divisor GCD and the smallest multiple LCM.

The least divisor is meaningless, since for any number it is always one. The greatest multiple is also meaningless, since the sequence of multiples goes to infinity.

Finding gcd

There are many methods for finding the greatest common divisor, the most famous of which are:

  • sequential search of divisors, selection of common ones for a pair and search for the largest of them;
  • decomposition of numbers into indivisible factors;
  • Euclidean algorithm;
  • binary algorithm.

Today at educational institutions The most popular are the methods of prime factorization and the Euclidean algorithm. The latter, in turn, is used when solving Diophantine equations: searching for GCD is required to check the equation for the possibility of resolution in integers.

Finding the NOC

The least common multiple is also determined by sequential search or decomposition into indivisible factors. In addition, it is easy to find the LCM if the greatest divisor has already been determined. For numbers X and Y, the LCM and GCD are related by the following relationship:

LCD(X,Y) = X × Y / GCD(X,Y).

For example, if GCM(15,18) = 3, then LCM(15,18) = 15 × 18 / 3 = 90. The most obvious example of using LCM is to find the common denominator, which is the least common multiple of given fractions.

Coprime numbers

If a pair of numbers has no common divisors, then such a pair is called coprime. The gcd for such pairs is always equal to one, and based on the connection between divisors and multiples, the gcd for coprime pairs is equal to their product. For example, the numbers 25 and 28 are relatively prime, because they have no common divisors, and LCM(25, 28) = 700, which corresponds to their product. Any two indivisible numbers will always be relatively prime.

Common divisor and multiple calculator

Using our calculator you can calculate GCD and LCM for an arbitrary number of numbers to choose from. Tasks on calculating common divisors and multiples are found in grades 5 and 6 arithmetic, but GCD and LCM are key concepts mathematics and are used in number theory, planimetry and communicative algebra.

Real life examples

Common denominator of fractions

Least common multiple is used when finding the common denominator of multiple fractions. Let's say in an arithmetic problem you need to sum 5 fractions:

1/8 + 1/9 + 1/12 + 1/15 + 1/18.

To add fractions, the expression must be reduced to a common denominator, which reduces to the problem of finding the LCM. To do this, select 5 numbers in the calculator and enter the values ​​of the denominators in the corresponding cells. The program will calculate the LCM (8, 9, 12, 15, 18) = 360. Now you need to calculate additional factors for each fraction, which are defined as the ratio of the LCM to the denominator. So the additional multipliers would look like:

  • 360/8 = 45
  • 360/9 = 40
  • 360/12 = 30
  • 360/15 = 24
  • 360/18 = 20.

After this, we multiply all the fractions by the corresponding additional factor and get:

45/360 + 40/360 + 30/360 + 24/360 + 20/360.

We can easily sum such fractions and get the result as 159/360. We reduce the fraction by 3 and see the final answer - 53/120.

Solving linear Diophantine equations

Linear Diophantine equations are expressions of the form ax + by = d. If the ratio d / gcd(a, b) is an integer, then the equation is solvable in integers. Let's check a couple of equations to see if they have an integer solution. First, let's check the equation 150x + 8y = 37. Using a calculator, we find GCD (150.8) = 2. Divide 37/2 = 18.5. The number is not an integer, therefore the equation does not have integer roots.

Let's check the equation 1320x + 1760y = 10120. Use a calculator to find GCD(1320, 1760) = 440. Divide 10120/440 = 23. As a result, we get an integer, therefore, the Diophantine equation is solvable in integer coefficients.

Conclusion

GCD and LCM play a big role in number theory, and the concepts themselves are widely used in a wide variety of areas of mathematics. Use our calculator to calculate greatest divisors and least multiples of any number of numbers.

The online calculator allows you to quickly find the greatest common divisor and least common multiple for two or any other number of numbers.

Calculator for finding GCD and LCM

Find GCD and LOC

Found GCD and LOC: 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 "Find GCD and LOC" button

How to enter numbers

  • Numbers are entered separated by a space, period or comma
  • The length of entered numbers is not limited, so find gcd and lcd long numbers it won't be difficult

What are GCD and NOC?

Greatest common divisor several numbers is the largest natural integer by which all original numbers are divisible without a remainder. The greatest common divisor is abbreviated as GCD.
Least common multiple several numbers is smallest number, which 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 properties of divisibility of numbers. Then, by combining them, you can check the divisibility of some of them and their combinations.

Some signs of divisibility of numbers

1. Divisibility test for 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 whether the number 34938 is divisible by 2.
Solution: We look at the last digit: 8 - that means the number is divisible by two.

2. Divisibility test for a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine whether a number is divisible by 3, you need to calculate the sum of the digits and check whether it is divisible by 3. Even if the sum of the digits is very large, you can repeat the same process again.
Example: determine whether the number 34938 is divisible by 3.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 3, which means the number is divisible by three.

3. Divisibility test for a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine whether the number 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.

4. Divisibility test for 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 whether the number 34938 is divisible by 9.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 9, which means 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 these numbers and select the largest of them.

Let's consider this method using the example of finding GCD(28, 36):

  1. We factor both numbers: 28 = 1·2·2·7, 36 = 1·2·2·3·3
  2. We find common factors, that is, those that both numbers have: 1, 2 and 2.
  3. 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 method is that you can write down the first multiples of two numbers, and then choose among them 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. Let's find the LCM for the same numbers 28 and 36:

  1. Find the product of numbers 28 and 36: 28·36 = 1008
  2. GCD(28, 36), as already known, is equal to 4
  3. LCM(28, 36) = 1008 / 4 = 252 .

Finding GCD and LCM for several numbers

The greatest common divisor can be found for several numbers, not just two. To do this, the numbers to be found for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. You can also use the following relation to find the gcd of several numbers: GCD(a, b, c) = GCD(GCD(a, b), c).

A similar relationship applies to the least common multiple: LCM(a, b, c) = LCM(LCM(a, b), c)

Example: find GCD and LCM for numbers 12, 32 and 36.

  1. First, let's factorize the numbers: 12 = 1·2·2·3, 32 = 1·2·2·2·2·2, 36 = 1·2·2·3·3.
  2. Let's find the common factors: 1, 2 and 2.
  3. Their product will give GCD: 1·2·2 = 4
  4. Now let’s find the LCM: to do this, let’s first find the LCM(12, 32): 12·32 / 4 = 96 .
  5. To find the LCM of all three numbers, you need to find GCD(96, 36): 96 = 1·2·2·2·2·2·3 , 36 = 1·2·2·3·3 , GCD = 1·2· 2 3 = 12.
  6. LCM(12, 32, 36) = 96·36 / 12 = 288.

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


If the numbers are large, find the common multiple of three or more numbers, then it is better to use another method of calculating the LCM.


To complete the task, you need to factor the given numbers into prime factors.


First you need to write down the decomposition of the largest number on a line, and below it - the rest.


In the expansion of each number there may be different quantity multipliers.


For example, let's factor the numbers 50 and 20 into prime factors.




In the expansion of the smaller number, you should highlight the factors that are missing in the expansion of the first largest number, and then add them to it. In the example presented, a two is missing.


Now you can calculate the least common multiple of 20 and 50.


LCM(20, 50) = 2 * 5 * 5 * 2 = 100


Thus, the product of the prime factors of the larger number and the factors of the second number that were not included in the expansion of the larger number will be the least common multiple.


To find the LCM of three or more numbers, you should factor them all into prime factors, as in the previous case.


As an example, you can find the least common multiple of the numbers 16, 24, 36.


36 = 2 * 2 * 3 * 3


24 = 2 * 2 * 2 * 3


16 = 2 * 2 * 2 * 2


Thus, only two twos from the expansion of sixteen were not included in the factorization of a larger number (one is in the expansion of twenty-four).


Thus, they need to be added to the expansion of a larger number.


LCM(12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9


There are special cases of determining the least common multiple. So, if one of the numbers can be divided without a remainder by another, then the larger of these numbers will be the least common multiple.


For example, the LCM of twelve and twenty-four is twenty-four.


If you need to find the least common multiple of each other prime numbers, which do not have identical divisors, then their LCM will be equal to their product.


For example, LCM (10, 11) = 110.

Mathematical expressions and problems require a lot of additional knowledge. NOC is one of the main ones, especially often used in The topic is studied in high school, and it is not particularly difficult to understand material; a person familiar with powers and the multiplication table will not have difficulty identifying the necessary numbers and discovering the result.

Definition

A common multiple is a number that can be completely divided into two numbers at the same time (a and b). Most often, this number is obtained by multiplying the original numbers a and b. The number must be divisible by both numbers at once, without deviations.

NOC is the short name adopted for the designation, collected from the first letters.

Ways to get a number

The method of multiplying numbers is not always suitable for finding the LCM; it is much better suited for simple single-digit or two-digit numbers. It is customary to divide into factors; the larger the number, the more factors there will be.

Example #1

For the simplest example, schools usually use prime, single- or double-digit numbers. For example, you need to solve the following task, find the least common multiple of the numbers 7 and 3, the solution is quite simple, just multiply them. As a result, there is a number 21, there is simply no smaller number.

Example No. 2

The second version of the task is much more difficult. The numbers 300 and 1260 are given, finding the LOC is mandatory. To solve the problem, the following actions are assumed:

Decomposition of the first and second numbers into simple factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7. The first stage is completed.

The second stage involves working with already obtained data. Each of the numbers received must participate in calculating the final result. For each multiplier, the most big number occurrences. NOC is total number, therefore, the factors from the numbers must be repeated in it, every single one, even those that are present in one copy. Both initial numbers contain the numbers 2, 3 and 5, in different degrees, 7 is present in only one case.

To calculate the final result, you need to take each number in the largest of the powers represented into the equation. All that remains is to multiply and get the answer; if filled out correctly, the task fits into two steps without explanation:

1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.

2) NOC = 6300.

That’s the whole problem, if you try to calculate the required number by multiplication, then the answer will definitely not be correct, since 300 * 1260 = 378,000.

Examination:

6300 / 300 = 21 - correct;

6300 / 1260 = 5 - correct.

The correctness of the result obtained is determined by checking - dividing the LCM by both original numbers; if the number is an integer in both cases, then the answer is correct.

What does NOC mean in mathematics?

As you know, there is not a single useless function in mathematics, this one is no exception. The most common purpose of this number is to reduce fractions to a common denominator. What is usually studied in grades 5-6 of secondary school. It is also additionally a common divisor for all multiples, if such conditions are present in the problem. Such an expression can find a multiple not only of two numbers, but also of a much larger number - three, five, and so on. How more numbers- the more actions there are in the task, but the complexity does not increase.

For example, given the numbers 250, 600 and 1500, you need to find their common LCM:

1) 250 = 25 * 10 = 5 2 *5 * 2 = 5 3 * 2 - this example describes factorization in detail, without reduction.

2) 600 = 60 * 10 = 3 * 2 3 *5 2 ;

3) 1500 = 15 * 100 = 33 * 5 3 *2 2 ;

In order to compose an expression, it is necessary to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is necessary to determine the maximum degree.

Attention: all factors must be brought to the point of complete simplification, if possible, decomposed to the level of single digits.

Examination:

1) 3000 / 250 = 12 - correct;

2) 3000 / 600 = 5 - true;

3) 3000 / 1500 = 2 - correct.

This method does not require any tricks or genius level abilities, everything is simple and clear.

Another way

In mathematics, many things are connected, many things can be solved in two or more ways, the same goes for finding the least common multiple, LCM. The following method can be used in the case of simple two-digit and single digit numbers. A table is compiled into which the multiplicand is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table using a line, take a number and write down the results of multiplying this number by integers, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers undergo the same computational process. Everything happens until a common multiple is found.

Given the numbers 30, 35, 42, you need to find the LCM connecting all the numbers:

1) Multiples of 30: 60, 90, 120, 150, 180, 210, 250, etc.

2) Multiples of 35: 70, 105, 140, 175, 210, 245, etc.

3) Multiples of 42: 84, 126, 168, 210, 252, etc.

It is noticeable that all the numbers are quite different, the only common number among them is 210, so it will be the NOC. Among the processes involved in this calculation there is also a greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but quite significant, LCM involves calculating the number that is divided by all given initial values, and GCD involves calculating the largest value by which the original numbers are divided.

The topic “Multiple Numbers” is studied in the 5th grade of secondary school. Its goal is to improve written and oral mathematical calculation skills. In this lesson, new concepts are introduced - “multiple numbers” and “divisors”, the technique of finding divisors and multiples of a natural number, and the ability to find LCM in various ways are practiced.

This topic is very important. Knowledge of 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.

Every natural number has an infinite number of multiples of it. It is itself considered the smallest. The multiple cannot be less than the number itself.

You need to prove that the number 125 is a multiple of 5. To do this, you need to 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 LOC.

1. If you need to find a common multiple of 2 numbers (for example, 80 and 20), where one of them (80) is divisible by the other (20), then this number (80) is the least 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 look at the last example. 6 and 7 in relation to 42 are divisors. They divide a multiple of a number without a remainder.

In this example, 6 and 7 are paired factors. Their product is equal to the most multiple 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.

Another example involves determining whether 9 is a divisor of 42.

42:9=4 (remainder 6)

Answer: 9 is not a divisor of 42 because the answer has a remainder.

A divisor differs from a multiple in that the divisor is the number that is divided by integers, and the multiple is itself divisible by this number.

Greatest common divisor of numbers a And b, multiplied by their least multiple, will give the product of the numbers themselves a And b.

Namely: gcd (a, b) x gcd (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 factor these numbers into prime factors and write them as a product of powers:

168=2³x3¹x7¹

2⁴х3³х5¹х7¹=15120

LCM(168, 180, 3024) = 15120.

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