Tool to calculate LCM. The lowest common multiple of two integers a and b is the smallest integer that is multiple of these two numbers.

LCM (Lowest Common Multiple) - dCode

Tag(s) : Arithmetics

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LCM is short for *Least Common Multiple* of 2 (or more) numbers. As its name suggests, for two (nonzero) integers $ a $ and $ b $, the LCM is the smallest (strictly positive) integer that is both a multiple of $ a $ and a multiple of $ b $.

**Method 1: list all multiples** and find the lowest common multiple.

__Example:__ LCM for `10` and `12``10` has these multiples: `0,10,20,30,40,50,60,70,etc.``12` has these multiples: `0,12,24,36,48,60,72,etc.`

The lowest common multiple is `60`.

**Method 2: use the prime factors decomposition**. The LCM is the multiplication of common factors by non-common factors

__Example:__ $ 10 = 2 \times 5 $ and $ 12 = 2 \times 2 \times 3 $

Common factors: `2` and non common factors: `2,3,5`

LCM(10, 12) = $ 2 \times 2 \times 3 \times 5 = 60 $

**Method 3: use the GCD value** and apply the formula `LCM(a, b) = a * b / GCD(a, b)`

__Example:__ GCD(10, 12) = 2

LCM(10, 12) = (10 * 12) / 2 = 60

**Method 1: list all multiples** and find the lowest common multiple.

__Example:__ LCM for 10, 12 and 15

10 has for multiples 0,10,20,30,40,50,60,70 etc.

12 has for multiples 0,12,24,36,48,60,72 etc.

15 has for multiples 0,15,30,45,60,75 etc.

The lowest common multiple is 60.

**Method 2: apply the LCM by 2** and use the formula `LCM(a,b,c) = LCM( LCM(a,b), c)`

__Example:__ LCM(10, 12) = 60

LCM(10, 12, 15) = LCM ( LCM(10, 12) , 15 ) = LCM(60,15) = 60

To calculate fractions and/or set fractions with the same denominator, calculate the lowest common multiple of the denominators (the fraction below the fraction line).

__Example:__ The fractions 7/8 and 15/36, their smallest common denominator is LCM(8,36)=72.

7/8 can therefore be written as 63/72 and 15/36 can be written 30/72.

Calculators has generally a function for LCM, else with GCD function, apply the formula:

$$ \text{L C M}(a, b) = \frac{ a \times b} { \text{G C D}(a, b) } $$

0 has no multiple, because no number can be divided by zero

LCM as it is mathematically defined, has no sense with non integers. However, it is possible to use this formula: CM(a*c,b*c) = CM(a,b)*c where CM is a common multiple (not the lowest) other rational numbers.

__Example:__ CM(1.2,2.4) = CM(12,24)/10 = 2

The following numbers have the property of having many divisors, some of them are highly composite numbers.

LCM(1,2,3)= | 6 |

LCM(1,2,3,4)= | 12 |

LCM(1,2,3,4,5)= | 60 |

LCM(1,2,3,4,5,6)= | 60 |

LCM(1,2,3…6,7)= | 420 |

LCM(1,2,3…7,8)= | 840 |

LCM(1,2,3…8,9)= | 2520 |

LCM(1,2,3…9,10)= | 2520 |

LCM(1,2,3…10,11)= | 27720 |

LCM(1,2,3…11,12)= | 27720 |

LCM(1,2,3…12,13)= | 360360 |

LCM(1,2,3…13,14)= | 360360 |

LCM(1,2,3…14,15)= | 360360 |

LCM(1,2,3…15,16)= | 720720 |

LCM(1,2,3…16,17)= | 12252240 |

LCM(1,2,3…17,18)= | 12252240 |

LCM(1,2,3…18,19)= | 232792560 |

LCM(1,2,3…19,20)= | 232792560 |

LCM(1,2,3…20,21)= | 232792560 |

LCM(1,2,3…21,22)= | 232792560 |

LCM(1,2,3…22,23)= | 5354228880 |

LCM(1,2,3…23,24)= | 5354228880 |

LCM(1,2,3…24,25)= | 26771144400 |

LCM(1,2,3…25,26)= | 26771144400 |

LCM(1,2,3…26,27)= | 80313433200 |

LCM(1,2,3…27,28)= | 80313433200 |

LCM(1,2,3…28,29)= | 2329089562800 |

LCM(1,2,3…29,30)= | 2329089562800 |

LCM(1,2,3…30,31)= | 72201776446800 |

LCM(1,2,3…31,32)= | 144403552893600 |

LCM(1,2,3…32,33)= | 144403552893600 |

LCM(1,2,3…33,34)= | 144403552893600 |

LCM(1,2,3…34,35)= | 144403552893600 |

LCM(1,2,3…35,36)= | 144403552893600 |

LCM(1,2,3…36,37)= | 5342931457063200 |

LCM(1,2,3…37,38)= | 5342931457063200 |

LCM(1,2,3…38,39)= | 5342931457063200 |

LCM(1,2,3…39,40)= | 5342931457063200 |

LCM(1,2,3…40,41)= | 219060189739591200 |

LCM(1,2,3…41,42)= | 219060189739591200 |

LCM(1,2,3…42,43)= | 9419588158802421600 |

LCM(1,2,3…43,44)= | 9419588158802421600 |

LCM(1,2,3…44,45)= | 9419588158802421600 |

LCM(1,2,3…45,46)= | 9419588158802421600 |

LCM(1,2,3…46,47)= | 442720643463713815200 |

LCM(1,2,3…47,48)= | 442720643463713815200 |

LCM(1,2,3…48,49)= | 3099044504245996706400 |

For any couple of 2 consecutive numbers, one is even and the other is odd, so only one is a multiple of 2. According to the method of computation of the LCM via the decomposition in prime factors, then the LCM is necessarily multiple of 2 which is a not common factor for the 2 numbers.

For any triplet of 3 consecutive numbers, only one is multiple of 3. According to the method of computation of the LCM via the decomposition in prime factors, then the LCM is necessarily multiple of 3 which is a not common factor for the 3 numbers.

The LCM is a common multiple of the 2 numbers, which is therefore a larger number having for divider the 2 numbers.

The GCD is a common divisor of the 2 numbers, which is therefore a smaller number having for multiple the 2 numbers.

The LCM and the CGD are linked by the formula: $$ \text{L C M}(a, b) = \frac{a \times b} { \text{G C D}(a, b) } $$

PPCM is a number that is a multiple of many, and it's as small as possible. This gives it a lot of mathematical advantage and simplifies the calculations.

__Example:__ A circle has 360° because 360 is divisible by 1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360 which is very practical.

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*LCM (Lowest Common Multiple)* on dCode.fr [online website], retrieved on 2023-10-01,

- LCM Calculator
- What is the LCM? (Définition)
- How to calculate the LCM? (Algorithm)
- How to calculate the LCM with multiple numbers? (LCM of 2 numbers or more)
- How to calculate the lowest common denominator of fractions?
- How to calculate LCM with a calculator (TI or Casio)?
- How to calculate LCM with a zero 0?
- How to calculate LCM with non-integers?
- What are LCM for the N first integers?
- Why the LCM of 2 consecutive numbers is a multiple of 2?
- Why the LCM of 3 consecutive numbers is a multiple of 3?
- What is the difference between LCM and Find the GDC to deduce the LCM?
- Why calculate the LCM?

lcm,gcd,lowest,smallest,common,multiple,divisor,algorithm,fraction,integer

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