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? (DĂ©finition)
The LCM (Least Common Multiple) is the smallest strictly positive integer that is a multiple of two or more non-zero integers.
For two (non-zero) integers $ a $ and $ b $, the LCM is therefore the smallest (strictly positive) integer that is both a multiple of $ a $ and a multiple of $ b $, in other words, the smallest number divisible by both $ a $ and $ 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,\dots $
$ 12 $ has these multiples: $ 0,12,24,36,48,60,72,\dots $
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 $
Thus $ \operatorname{LCM}(10, 12) = 2 \times 2 \times 3 \times 5 = 60 $
— Method 3: use the GCD value and apply the formula $$ \operatorname{LCM}(a,b) = \frac{ a \times b } { \operatorname{GCD}(a,b) } $$
Example: $ \operatorname{GCD}(10,12) = 2 \\ \operatorname{LCM}(10, 12) = (10 \times 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,\dots $
$ 12 $ has for multiples $ 0,12,24,36,48,60,72,\dots $
$ 15 $ has for multiples $ 0,15,30,45,60,75,\dots $
The lowest common multiple is $ 60 $.
Method 2: apply the LCM by pairs and use the formula $$ \operatorname{LCM}(a,b,c) = \operatorname{LCM}( \operatorname{LCM}(a,b), c) $$
Example: $ \operatorname{LCM}(10, 12) = 60 \\ \operatorname{LCM}(10, 12, 15) = \operatorname{LCM}( \operatorname{LCM}(10, 12) , 15 ) = \operatorname{LCM}(60,15) = 60 $
How to calculate the lowest common denominator of fractions?
To reduce fractions to a common denominator, calculate the LCM of the denominators (the part below the fraction bar).
Example: The fractions $ 7/8 $ and $ 15/36 $, their smallest common denominator is $ \operatorname{PPCM}(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:
$$ \operatorname{LCM}(a, b) = \frac{ a \times b} { \operatorname{GCD}(a, b) } $$
How to calculate LCM with a zero 0?
The LCM is not defined if one of the numbers is zero (equal to zero), because no number can be divided by 0.
How to calculate LCM with non-integers?
The least common multiple (LCM) is defined only for 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 necessarily the smallest) of the rational numbers.
Thus, for decimal numbers, multiply each number by a power of 10 until you obtain integers, calculate the LCM, and then divide the result by that power.
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.
How do you calculate the LCM of coprime 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: $$ \operatorname{LCM}(a, b) = \frac{a \times b} { \operatorname{GCD}(a, b) } $$
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? (DĂ©finition)
- 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?
- How do you calculate the LCM of coprime numbers?
- What is the difference between LCM and Find the GDC to deduce the LCM?
- Why calculate the LCM?
