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 (Lowest Common Multiple)
LCM Calculator
Answers to Questions (FAQ)
What is the LCM? (Définition)
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 $.
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 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
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.
What is the difference between LCM and GCD?
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: $$ \text{L C M}(a, b) = \frac{a \times b} { \text{G C D}(a, b) } $$
Why calculate the LCM?
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
- 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?
