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LCM (Lowest Common Multiple)

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.

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LCM (Lowest Common Multiple) -

Tag(s) : Arithmetics

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# LCM (Lowest Common Multiple)

## Calculus of LCM of fractions

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.

### How to calculate the LCM? (Algorithm)

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 * 2 * 3 * 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

### How to calculate the LCM with multiple numbers? (LCM of 2 numbers or more)

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

### How to calculate the lowest common denominator of fractions?

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.

### How to calculate LCM with a calculator (TI or Casio)?

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

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

### How to calculate LCM with a zero 0?

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

### How to calculate LCM with non-integers?

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

### What are LCM for the N first integers?

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

### Why the LCM of 2 consecutive numbers is a multiple of 2?

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.

### Why the LCM of 3 consecutive numbers is a multiple of 3?

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.

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