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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,Mathematics

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LCM (Lowest Common Multiple)

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Calculus of LCM of any numbers



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.

Answers to Questions

How to calculate the LCM? (Algorithm)

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

Let the numbers be 10 and 12, one wants to find the LCM
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 factorshref decomposition. The LCM is the multiplicationhref of common factors by non-commonhref factors

10 = 2 * 5
12 = 2 * 2 * 3
Common factors : 2
Non commonhref factors : 2, 3, 5
LCM(10, 12) = 2 * 2 * 3 * 5 = 60

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

GCDhref(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 multiples and find the lowest common multiple.

Let the numbers be 10 and 12, one wants to find the LCM
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: use the formula LCM(a,b,c) = LCM( LCM(a,b), c)

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 set fractions with the same denominator, you have to calculate the lowest common multiple of the denominators (the fraction below the fraction line).

Consider 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, if you only have GCDhref, apply the formula: $$ LCM(a, b) = a * b / GCDhref(a, b) $$

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.

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

What are LCM for the N first integers?

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

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