Online calculator allows you to quickly find the greatest common divisor and least common multiple of both two and 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 finding GCD and LCM of long numbers is not 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.

Definition. The largest natural number by which the numbers a and b are divided without remainder is called greatest common divisor (GCD) these numbers.

Let's find the greatest common divisor of the numbers 24 and 35.
The divisors of 24 are the numbers 1, 2, 3, 4, 6, 8, 12, 24, and the divisors of 35 are the numbers 1, 5, 7, 35.
We see that the numbers 24 and 35 have only one common divisor - the number 1. Such numbers are called mutually prime.

Definition. Natural numbers are called mutually prime, if their greatest common divisor (GCD) is 1.

Greatest Common Divisor (GCD) can be found without writing out all the divisors of the given numbers.

Let's factor the numbers 48 and 36 and get:
48 = 2 * 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3.
From the factors included in the expansion of the first of these numbers, we cross out those that are not included in the expansion of the second number (i.e., two twos).
The factors remaining are 2 * 2 * 3. Their product is equal to 12. This number is the greatest common divisor of the numbers 48 and 36. The greatest common divisor of three or more numbers is also found.

To find greatest common divisor

2) from the factors included in the expansion of one of these numbers, cross out those that are not included in the expansion of other numbers;
3) find the product of the remaining factors.

If all given numbers are divisible by one of them, then this number is greatest common divisor given numbers.
For example, the greatest common divisor of the numbers 15, 45, 75 and 180 is the number 15, since all other numbers are divisible by it: 45, 75 and 180.

Least common multiple (LCM)

Definition. Least common multiple (LCM) natural numbers a and b are the smallest natural number that is a multiple of both a and b. The least common multiple (LCM) of the numbers 75 and 60 can be found without writing down the multiples of these numbers in a row. To do this, let's factor 75 and 60 into prime factors: 75 = 3 * 5 * 5, and 60 = 2 * 2 * 3 * 5.
Let's write down the factors included in the expansion of the first of these numbers, and add to them the missing factors 2 and 2 from the expansion of the second number (i.e., we combine the factors).
We get five factors 2 * 2 * 3 * 5 * 5, the product of which is 300. This number is the least common multiple of the numbers 75 and 60.

They also find the least common multiple of three or more numbers.

To find least common multiple several natural numbers, you need:
1) factor them into prime factors;
2) write down the factors included in the expansion of one of the numbers;
3) add to them the missing factors from the expansions of the remaining numbers;
4) find the product of the resulting factors.

Note that if one of these numbers is divisible by all other numbers, then this number is the least common multiple of these numbers.
For example, the least common multiple of the numbers 12, 15, 20, and 60 is 60 because it is divisible by all of those numbers.

Pythagoras (VI century BC) and his students studied the question of the divisibility of numbers. They called a number equal to the sum of all its divisors (without the number itself) a perfect number. For example, the numbers 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The next perfect numbers are 496, 8128, 33,550,336. The Pythagoreans only knew the first three perfect numbers. The fourth - 8128 - became known in the 1st century. n. e. The fifth - 33,550,336 - was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there are odd perfect numbers or whether there is a largest perfect number.
The interest of ancient mathematicians in prime numbers is due to the fact that any number is either prime or can be represented as a product of prime numbers, i.e. prime numbers are like bricks from which the rest of the natural numbers are built.
You probably noticed that prime numbers in the series of natural numbers occur unevenly - in some parts of the series there are more of them, in others - less. But the further we move along the number series, the less common prime numbers are. The question arises: is there a last (largest) prime number? The ancient Greek mathematician Euclid (3rd century BC), in his book “Elements,” which was the main textbook of mathematics for two thousand years, proved that there are infinitely many prime numbers, i.e. behind every prime number there is an even greater prime number.
To find prime numbers, another Greek mathematician of the same time, Eratosthenes, came up with this method. He wrote down all the numbers from 1 to some number, and then crossed out one, which is neither a prime nor a composite number, then crossed out through one all the numbers coming after 2 (numbers that are multiples of 2, i.e. 4, 6 , 8, etc.). The first remaining number after 2 was 3. Then, after two, all numbers coming after 3 (numbers that were multiples of 3, i.e. 6, 9, 12, etc.) were crossed out. in the end only the prime numbers remained uncrossed.

To reduce fractions to the smallest common denominator, you need to: 1) find the least common multiple of the denominators of these fractions, it will be the least common denominator. 2) find an additional factor for each fraction, why divide new denominator to the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.

Examples. Reduce the following fractions to their lowest common denominator.

We find the least common multiple of the denominators: LCM(5; 4) = 20, since 20 is the smallest number that is divisible by both 5 and 4. Find for the 1st fraction an additional factor 4 (20 : 5=4). For the 2nd fraction the additional factor is 5 (20 : 4=5). We multiply the numerator and denominator of the 1st fraction by 4, and the numerator and denominator of the 2nd fraction by 5. We have reduced these fractions to the lowest common denominator ( 20 ).

The lowest common denominator of these fractions is the number 8, since 8 is divisible by 4 and itself. There will be no additional factor for the 1st fraction (or we can say that it is equal to one), for the 2nd fraction the additional factor is 2 (8 : 4=2). We multiply the numerator and denominator of the 2nd fraction by 2. We have reduced these fractions to the lowest common denominator ( 8 ).

These fractions are not irreducible.

Let's reduce the 1st fraction by 4, and reduce the 2nd fraction by 2. ( see examples on reducing ordinary fractions: Sitemap → 5.4.2. Examples of reducing common fractions). Find the LOC(16 ; 20)=2 4 · 5=16· 5=80. The additional multiplier for the 1st fraction is 5 (80 : 16=5). The additional factor for the 2nd fraction is 4 (80 : 20=4). We multiply the numerator and denominator of the 1st fraction by 5, and the numerator and denominator of the 2nd fraction by 4. We have reduced these fractions to the lowest common denominator ( 80 ).

We find the lowest common denominator NCD(5 ; 6 and 15)=NOK(5 ; 6 and 15)=30. The additional factor to the 1st fraction is 6 (30 : 5=6), the additional factor to the 2nd fraction is 5 (30 : 6=5), the additional factor to the 3rd fraction is 2 (30 : 15=2). We multiply the numerator and denominator of the 1st fraction by 6, the numerator and denominator of the 2nd fraction by 5, the numerator and denominator of the 3rd fraction by 2. We have reduced these fractions to the lowest common denominator ( 30 ).

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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 large number occurrences. The LCM is a general number, so the factors of 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 powers; 7 is present only in 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 high school. It is also additionally a common divisor for all multiples, if such conditions are present in the problem. Such an expression can find multiples not only of two numbers, but also of much larger numbers - 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 a number that is divided by all given initial values, and GCD involves calculating highest value by which the original numbers are divided.

To solve examples with fractions, you need to be able to find the lowest common denominator. Below are detailed instructions.

How to find the lowest common denominator - concept

Least common denominator (LCD) in simple words is the minimum number that is divisible by the denominators of all fractions in this example. In other words, it is called the Least Common Multiple (LCM). NOS is used only if the denominators of the fractions are different.

How to find the lowest common denominator - examples

Let's look at examples of finding NOCs.

Calculate: 3/5 + 2/15.

Solution (Sequence of actions):

• We look at the denominators of the fractions, make sure that they are different and that the expressions are as abbreviated as possible.
• We find the smallest number that is divisible by both 5 and 15. This number will be 15. Thus, 3/5 + 2/15 = ?/15.
• We figured out the denominator. What will be in the numerator? An additional multiplier will help us figure this out. An additional factor is the number obtained by dividing the NZ by the denominator of a particular fraction. For 3/5, the additional factor is 3, since 15/5 = 3. For the second fraction, the additional factor is 1, since 15/15 = 1.
• Having found out the additional factor, we multiply it by the numerators of the fractions and add the resulting values. 3/5 + 2/15 = (3*3+2*1)/15 = (9+2)/15 = 11/15.

Answer: 3/5 + 2/15 = 11/15.

If in the example we add or subtract not 2, but 3 or more fractions, then the NCD must be looked for for as many fractions as are given.

Calculate: 1/2 – 5/12 + 3/6

Solution (sequence of actions):

• Finding the lowest common denominator. The minimum number divisible by 2, 12 and 6 is 12.
• We get: 1/2 – 5/12 + 3/6 = ?/12.
• We are looking for additional multipliers. For 1/2 – 6; for 5/12 – 1; for 3/6 – 2.
• We multiply by the numerators and assign the corresponding signs: 1/2 – 5/12 + 3/6 = (1*6 – 5*1 + 2*3)/12 = 7/12.

Answer: 1/2 – 5/12 + 3/6 = 7/12.