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Common multiple of three numbers. School of mathematics for everyone who studies and teaches

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: 11074

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 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 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

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'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.

Finding the NOC

In order to find common denominator When adding and subtracting fractions with different denominators, you must know and be able to calculate least common multiple (LCM).

A multiple of a is a number that is itself divisible by a without a remainder.
Numbers that are multiples of 8 (that is, these numbers are divisible 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. There is a finite number of divisors.

A common multiple of two natural numbers is a number that is divisible by both of these numbers.

  • The 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 NOC
LCM can be found and written in two ways.

The first way to find the LOC
This method is usually used for small numbers.
1. Write down the multiples for each number on a line until you find a multiple that is the same for both numbers.
2. A multiple of a is denoted by the capital letter “K”.

K(a) = (...,...)
Example. Find LOC 6 and 8.
K (6) = (12, 18, 24, 30, ...)

K(8) = (8, 16, 24, 32, ...)

LCM(6, 8) = 24

The second way to find the LOC
This method is convenient to use to find the LCM for three or more numbers.
1. Divide the given numbers into simple multipliers. You can read more about the rules for factoring into prime factors in the topic of how to find the greatest common divisor (GCD).


2. Write down the factors included in the expansion on a line the biggest of numbers, and below it is the decomposition of the remaining numbers.

  • The number of identical factors in decompositions of numbers can be different.

60 = 2 . 2 . 3 . 5

24 = 2 . 2 . 2 . 3
3. Emphasize in decomposition less numbers (smaller numbers) factors that were not included in the expansion of the larger number (in our example it is 2) and add these factors to the expansion of the larger number.
LCM(24, 60) = 2. 2. 3. 5. 2
4. Write down the resulting product as an answer.
Answer: LCM (24, 60) = 120

You can also formalize finding the least common multiple (LCM) as follows. Let's find the LOC (12, 16, 24).


24 = 2 . 2 . 2 . 3

16 = 2 . 2 . 2 . 2

12 = 2 . 2 . 3

As we see from the decomposition of numbers, all factors of 12 are included in the decomposition of 24 (the largest of the numbers), so we add only one 2 from the decomposition 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 an NOC
1. If one of the numbers is divisible by the others, then the least common multiple of these numbers is equal to this number.
For example, LCM (60, 15) = 60
2. Since relatively prime numbers do not have common prime factors, their least common multiple is equal to the product of these numbers.
Example.
LCM(8, 9) = 72

Let's find the greatest common divisor of GCD (36; 24)

Solution steps

Method No. 1

36 - composite number
24 - composite number

Let's expand the number 36

36: 2 = 18
18: 2 = 9 - divisible by the prime number 2
9: 3 = 3 - divisible by the prime number 3.

Let's break down the number 24 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

24: 2 = 12 - divisible by the prime number 2
12: 2 = 6 - divisible by the prime number 2
6: 2 = 3
We complete the division since 3 is a prime number

2) Highlight it in blue and write out the common factors

36 = 2 ⋅ 2 ⋅ 3 ⋅ 3
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
Common factors (36; 24): 2, 2, 3

3) Now, to find the GCD you need to multiply the common factors

Answer: GCD (36; 24) = 2 ∙ 2 ∙ 3 ​​= 12

Method No. 2

1) Find all possible divisors of the numbers (36; 24). To do this, we will alternately divide the number 36 into divisors from 1 to 36, and the number 24 into divisors from 1 to 24. If the number is divisible without a remainder, then we write the divisor in the list of divisors.

For number 36
36: 1 = 36; 36: 2 = 18; 36: 3 = 12; 36: 4 = 9; 36: 6 = 6; 36: 9 = 4; 36: 12 = 3; 36: 18 = 2; 36: 36 = 1;

For the number 24 Let's write down all the cases when it is divisible without a remainder:
24: 1 = 24; 24: 2 = 12; 24: 3 = 8; 24: 4 = 6; 24: 6 = 4; 24: 8 = 3; 24: 12 = 2; 24: 24 = 1;

2) Let’s write down all the common divisors of the numbers (36; 24) and highlight the largest one in green, this will be the greatest common divisor of the gcd of the numbers (36; 24)

Common factors of numbers (36; 24): 1, 2, 3, 4, 6, 12

Answer: GCD (36 ; 24) = 12



Let's find the least common multiple of the LCM (52; 49)

Solution steps

Method No. 1

1) Let's factor the numbers into prime factors. To do this, let’s check whether each of the numbers is prime (if a number is prime, then it cannot be decomposed into prime factors, and it is itself a decomposition)

52 - composite number
49 - composite number

Let's expand the number 52 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

52: 2 = 26 - divisible by the prime number 2
26: 2 = 13 - divisible by the prime number 2.
We complete the division since 13 is a prime number

Let's expand the number 49 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

49: 7 = 7 - divisible by the prime number 7.
We complete the division since 7 is a prime number

2) First of all, write down the factors of the largest number, and then the smaller number. Let's find the missing factors, highlight in blue in the expansion of the smaller number the factors that were not included in the expansion of the larger number.

52 = 2 ∙ 2 ∙ 13
49 = 7 ∙ 7

3) Now, to find the LCM you need to multiply the factors of the larger number with the missing factors, which are highlighted in blue

LCM (52 ; 49) = 2 ∙ 2 ∙ 13 ∙ 7 ∙ 7 = 2548

Method No. 2

1) Find all possible multiples of the numbers (52; 49). To do this, we will alternately multiply the number 52 by the numbers from 1 to 49, and the number 49 by the numbers from 1 to 52.

Select all multiples 52 in green:

52 ∙ 1 = 52 ; 52 ∙ 2 = 104 ; 52 ∙ 3 = 156 ; 52 ∙ 4 = 208 ;
52 ∙ 5 = 260 ; 52 ∙ 6 = 312 ; 52 ∙ 7 = 364 ; 52 ∙ 8 = 416 ;
52 ∙ 9 = 468 ; 52 ∙ 10 = 520 ; 52 ∙ 11 = 572 ; 52 ∙ 12 = 624 ;
52 ∙ 13 = 676 ; 52 ∙ 14 = 728 ; 52 ∙ 15 = 780 ; 52 ∙ 16 = 832 ;
52 ∙ 17 = 884 ; 52 ∙ 18 = 936 ; 52 ∙ 19 = 988 ; 52 ∙ 20 = 1040 ;
52 ∙ 21 = 1092 ; 52 ∙ 22 = 1144 ; 52 ∙ 23 = 1196 ; 52 ∙ 24 = 1248 ;
52 ∙ 25 = 1300 ; 52 ∙ 26 = 1352 ; 52 ∙ 27 = 1404 ; 52 ∙ 28 = 1456 ;
52 ∙ 29 = 1508 ; 52 ∙ 30 = 1560 ; 52 ∙ 31 = 1612 ; 52 ∙ 32 = 1664 ;
52 ∙ 33 = 1716 ; 52 ∙ 34 = 1768 ; 52 ∙ 35 = 1820 ; 52 ∙ 36 = 1872 ;
52 ∙ 37 = 1924 ; 52 ∙ 38 = 1976 ; 52 ∙ 39 = 2028 ; 52 ∙ 40 = 2080 ;
52 ∙ 41 = 2132 ; 52 ∙ 42 = 2184 ; 52 ∙ 43 = 2236 ; 52 ∙ 44 = 2288 ;
52 ∙ 45 = 2340 ; 52 ∙ 46 = 2392 ; 52 ∙ 47 = 2444 ; 52 ∙ 48 = 2496 ;
52 ∙ 49 = 2548 ;

Select all multiples 49 in green:

49 ∙ 1 = 49 ; 49 ∙ 2 = 98 ; 49 ∙ 3 = 147 ; 49 ∙ 4 = 196 ;
49 ∙ 5 = 245 ; 49 ∙ 6 = 294 ; 49 ∙ 7 = 343 ; 49 ∙ 8 = 392 ;
49 ∙ 9 = 441 ; 49 ∙ 10 = 490 ; 49 ∙ 11 = 539 ; 49 ∙ 12 = 588 ;
49 ∙ 13 = 637 ; 49 ∙ 14 = 686 ; 49 ∙ 15 = 735 ; 49 ∙ 16 = 784 ;
49 ∙ 17 = 833 ; 49 ∙ 18 = 882 ; 49 ∙ 19 = 931 ; 49 ∙ 20 = 980 ;
49 ∙ 21 = 1029 ; 49 ∙ 22 = 1078 ; 49 ∙ 23 = 1127 ; 49 ∙ 24 = 1176 ;
49 ∙ 25 = 1225 ; 49 ∙ 26 = 1274 ; 49 ∙ 27 = 1323 ; 49 ∙ 28 = 1372 ;
49 ∙ 29 = 1421 ; 49 ∙ 30 = 1470 ; 49 ∙ 31 = 1519 ; 49 ∙ 32 = 1568 ;
49 ∙ 33 = 1617 ; 49 ∙ 34 = 1666 ; 49 ∙ 35 = 1715 ; 49 ∙ 36 = 1764 ;
49 ∙ 37 = 1813 ; 49 ∙ 38 = 1862 ; 49 ∙ 39 = 1911 ; 49 ∙ 40 = 1960 ;
49 ∙ 41 = 2009 ; 49 ∙ 42 = 2058 ; 49 ∙ 43 = 2107 ; 49 ∙ 44 = 2156 ;
49 ∙ 45 = 2205 ; 49 ∙ 46 = 2254 ; 49 ∙ 47 = 2303 ; 49 ∙ 48 = 2352 ;
49 ∙ 49 = 2401 ; 49 ∙ 50 = 2450 ; 49 ∙ 51 = 2499 ; 49 ∙ 52 = 2548 ;

2) Let’s write down all the common multiples of the numbers (52; 49) and highlight the smallest one in green, this will be the smallest common multiple of the numbers (52; 49).

Common multiples of numbers (52; 49): 2548

Answer: LCM (52; 49) = 2548

The least common multiple of two numbers is directly related to the greatest common divisor of those numbers. This connection between GCD and NOC is determined by the following theorem.

Theorem.

The least common multiple of two positive integers a and b is equal to the product of a and b divided by the greatest common divisor of a and b, that is, LCM(a, b)=a b:GCD(a, b).

Proof.

Let M is some multiple of the numbers a and b. That is, M is divisible by a, and by the definition of divisibility, there is some integer k such that the equality M=a·k is true. But M is also divisible by b, then a·k is divisible by b.

Let's denote gcd(a, b) as d. Then we can write the equalities a=a 1 ·d and b=b 1 ·d, and a 1 =a:d and b 1 =b:d will be relatively prime numbers. Consequently, the condition obtained in the previous paragraph that a · k is divisible by b can be reformulated as follows: a 1 · d · k is divided by b 1 · d , and this, due to the properties of divisibility, is equivalent to the condition that a 1 · k is divisible by b 1.

You also need to write down two important corollaries from the theorem considered.

    The common multiples of two numbers are the same as the multiples of their least common multiple.

    This is indeed the case, since any common multiple of M of the numbers a and b is determined by the equality M=LMK(a, b)·t for some integer value t.

    The least common multiple of mutually prime positive numbers a and b is equal to their product.

    The rationale for this fact is quite obvious. Since a and b are relatively prime, then gcd(a, b)=1, therefore, GCD(a, b)=a b: GCD(a, b)=a b:1=a b.

Least common multiple of three or more numbers

Finding the least common multiple of three or more numbers can be reduced to sequentially finding the LCM of two numbers. How this is done is indicated in the following theorem. a 1 , a 2 , …, a k coincide with the common multiples of the numbers m k-1 and a k , therefore, coincide with the common multiples of the number m k . And since the smallest positive multiple of the number m k is the number m k itself, then the smallest common multiple of the numbers a 1, a 2, ..., a k is m k.

References.

  • Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
  • Vinogradov I.M. Fundamentals of number theory.
  • Mikhelovich Sh.H. Number theory.
  • Kulikov L.Ya. and others. Collection of problems in algebra and number theory: Textbook for students of physics and mathematics. specialties of pedagogical institutes.