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

LCM (Lowest Common Multiple) - dCode

Tag(s) : Arithmetics

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

__Example:__ **LCM** for 10 and 12

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.

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 Calculator
- 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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