# Least common multiple

The least common multiple ( LCM ) is a mathematical term. Its counterpart is the greatest common factor (GCF). Both play a role in fractions and number theory , among other things .

The least common multiple of two whole numbers and is the smallest positive natural number that is both a multiple of and a multiple of . In addition, for the case or the LCM is defined as . ${\ displaystyle m}$${\ displaystyle n}$ ${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle m = 0}$${\ displaystyle n = 0}$${\ displaystyle \ operatorname {kgV} (m, \, n): = 0}$

The English term for the smallest common multiple is least common multiple , or lcm for short, and is also used in mathematical texts.

## Calculation of the LCM of natural numbers

### Example of LCM calculation

• The positive multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, 108, ...
• The positive multiples of 18 are: 18, 36, 54, 72, 90, 108, ...
• The common positive multiples of 12 and 18 are therefore 36, 72, 108, ...
• and the smallest of these is 36; in characters:
${\ displaystyle \ operatorname {kgV} (12.18) = 36}$

### Calculation using prime factorization

GCF and LCM can be determined by the prime factorization of the two given numbers. Example:

${\ displaystyle 3528 = 2 ^ {\ color {Red} 3} \ cdot 3 ^ {\ color {Red} 2} \ cdot 7 ^ {\ color {Red} 2}}$
${\ displaystyle 3780 = 2 ^ {\ color {OliveGreen} 2} \ cdot 3 ^ {\ color {OliveGreen} 3} \ cdot 5 ^ {\ color {OliveGreen} 1} \ cdot 7 ^ {\ color {OliveGreen} 1 }}$

For the LCM one takes the prime factors that occur in at least one of the two decompositions, and as the associated exponent, the larger of the initial exponents:

${\ displaystyle \ operatorname {kgV} (3528,3780) = 2 ^ {\ color {Red} 3} \ cdot 3 ^ {\ color {OliveGreen} 3} \ cdot 5 ^ {\ color {OliveGreen} 1} \ cdot 7 ^ {\ color {Red} 2} = 52,920}$

### Calculation using the greatest common divisor (GCD)

The following formula applies:

${\ displaystyle \ mathrm {ggT} (m, n) \ cdot \ mathrm {kgV} (m, n) = | m \ cdot n |}$

If both numbers are positive or negative, the amount bars are omitted. This can be used to calculate this if the has already been determined (e.g. with the Euclidean algorithm ) (conversely, this formula can also be used to calculate the one from the ). ${\ displaystyle \ mathrm {kgV}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {kgV}}$

After determining one of the two numbers , it is usually easiest to divide by the and multiply by the other number. The amount of the result is what you are looking for . Example: ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {kgV}}$

The 24 and 18 is 6 (to calculate the means Euclidean algorithm see the article to GCD ). This is consequently (since both numbers are positive, the amount is omitted) ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {kgV}}$

${\ displaystyle (18 \ div 6) \ cdot 24 = 3 \ cdot 24 = 72}$.

The formula at the beginning of the section is easy to understand, by the way, because the product of the numbers can also be expressed as follows: ${\ displaystyle \ mathrm {ggT}}$

${\ displaystyle m \ cdot n = a \ cdot \ mathrm {ggT} (m, n) \ cdot b \ cdot \ mathrm {ggT} (m, n)}$.

Now this can be determined with the using the factors and , since these are coprime and thus their product with which gives the smallest common multiple (the amount is necessary if one of the two numbers is negative): ${\ displaystyle \ mathrm {kgV}}$${\ displaystyle \ mathrm {ggT}}$${\ textstyle a}$${\ textstyle b}$${\ displaystyle \ mathrm {ggT}}$

${\ displaystyle \ mathrm {kgV} (m, n) = | a \ cdot b \ cdot \ mathrm {ggT} (m, n) |}$

Multiplying both sides by the and using the relationship of the previous equation results in the first equation of the section. ${\ displaystyle \ mathrm {ggT}}$

${\ displaystyle \ mathrm {ggT} (m, n) \ cdot \ mathrm {kgV} (m, n) = \ mathrm {ggT} (m, n) \ cdot | a \ cdot b \ cdot \ mathrm {ggT} (m, n) | = | m \ cdot n |}$

## The LCM of several numbers

All prime factors that occur in at least one of the numbers are used with the highest power that occurs, for example:

${\ displaystyle 144 = 2 ^ {\ color {Red} 4} \ cdot 3 ^ {\ color {Red} 2}}$
${\ displaystyle 160 = 2 ^ {\ color {OliveGreen} 5} \ cdot 5 ^ {\ color {OliveGreen} 1}}$
${\ displaystyle 175 = 5 ^ {\ color {Blue} 2} \ cdot 7 ^ {\ color {Blue} 1},}$

so:

${\ displaystyle \ mathrm {kgV} (144,160,175) = 2 ^ {\ color {OliveGreen} 5} \ cdot 3 ^ {\ color {Red} 2} \ cdot 5 ^ {\ color {Blue} 2} \ cdot 7 ^ {\ color {Blue} 1} = 50,400.}$

You could also calculate first and then as a two-digit combination on the whole numbers, this is associative : ${\ displaystyle \ mathrm {kgV} (144,160) = 1440}$${\ displaystyle \ mathrm {kgV} (1440.175) = 50,400,}$${\ displaystyle \ mathrm {kgV}}$

${\ displaystyle \ mathrm {kgV} (m, \, \ mathrm {kgV} (n, p)) = \ mathrm {kgV} (\ mathrm {kgV} (m, n), \, p).}$

This justifies the spelling ${\ displaystyle \ mathrm {kgV} (m, n, p)}$

## Applications

### Fractions

Suppose you want to add the fractions and . To do this, they must be brought to a common denominator by expanding . It could be with Multiply what gives. The lowest possible common denominator (the so-called main denominator ) is . The two fractions are expanded to this denominator and then added: ${\ displaystyle {\ tfrac {17} {21}}}$${\ displaystyle {\ tfrac {44} {35}}}$${\ displaystyle 21}$${\ displaystyle 35}$${\ displaystyle 735}$${\ displaystyle \ operatorname {kgV} (21.35) = 105}$

${\ displaystyle {\ frac {17} {21}} + {\ frac {44} {35}} = {\ frac {{\ color {Red} 5} \ cdot 17} {{\ color {Red} 5} \ cdot 21}} + {\ frac {{\ color {Red} 3} \ cdot 44} {{\ color {Red} 3} \ cdot 35}} = {\ frac {85} {105}} + {\ frac {132} {105}} = {\ frac {217} {105}}}$

## The LCM in rings

This is defined analogously to in rings : A ring element is called the smallest common multiple of two ring elements and if is a common multiple of and and in turn every other common multiple of and is a multiple of . ${\ displaystyle \ operatorname {ggT}}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle v}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle v}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle v}$

Formally, this definition for a ring is written like this: ${\ displaystyle R}$

${\ displaystyle v = \ mathrm {kgV} (a, b) \ quad: \ Longleftrightarrow \ quad a \ mid v, \; b \ mid v, \; \ forall e \ in R: (a \ mid e, \ , b \ mid e) \ Rightarrow v \ mid e}$

This more general definition can be extended to several numbers (even to an infinite number).

### Examples

#### The LCM of polynomials

This cannot only be defined for natural (and whole) numbers. You can z. B. also form for polynomials . Instead of the prime factorization , one takes here the decomposition into irreducible factors: ${\ displaystyle \ mathrm {kgV}}$

{\ displaystyle {\ begin {aligned} f (x) & = x ^ {2} + 2xy + y ^ {2} = (x + y) ^ {2} \\ g (x) & = x ^ {2 } -y ^ {2} = (x + y) (xy) \ end {aligned}}}

Then

${\ displaystyle \ mathrm {kgV} (f, g) = (x + y) ^ {2} (xy)}$.

The remainder division that exists also for polynomials, the identification of, common divisors.

#### Gaussian number ring

In the Gaussian number ring , the greatest common divisor of and is even , because and . Strictly speaking, is a greatest common factor, since all the numbers associated with this number are also greatest common factors. ${\ displaystyle \ mathbb {Z} + \ mathrm {i} \ mathbb {Z}}$${\ displaystyle 2}$${\ displaystyle 1 + 3 \ mathrm {i}}$${\ displaystyle 1+ \ mathrm {i}}$${\ displaystyle 2 = - \ mathrm {i} (1+ \ mathrm {i}) ^ {2}}$${\ displaystyle 1 + 3 \ mathrm {i} = (1+ \ mathrm {i}) (2+ \ mathrm {i})}$${\ displaystyle 1+ \ mathrm {i}}$

Not in every ring one or one exists for two elements. If they have one, they can have several . If the ring is an integrity ring , then all are associated with one another , in signs . ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ sim}$

If an integrity ring is and the elements and have one , then they have one too , and the equation applies ${\ displaystyle R}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle \ mathrm {ggT}}$

${\ displaystyle a \ cdot b \ sim \ mathrm {ggT} (a, b) \ cdot \ mathrm {kgV} (a, b)}$

However, if only one of and exists, then there does not necessarily have to be one . ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle \ mathrm {kgV}}$

##### Integrity ring

In the integrity ring have the elements ${\ displaystyle R = \ mathbb {Z} [{\ sqrt {-3}}]}$

${\ displaystyle a: = 4 = 2 \ cdot 2 = (1 + {\ sqrt {-3}}) (1 - {\ sqrt {-3}}), \ quad b: = (1 + {\ sqrt { -3}}) \ cdot 2}$

none : The elements and are two maximum common factors , because both have the same amount . However, these two elements are not associated with each other, so there is no of and . ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle 1 + {\ sqrt {-3}}}$${\ displaystyle 2}$${\ displaystyle \ mathrm {ggT}}$${\ displaystyle a}$${\ displaystyle b}$

However, the elements mentioned and have their own GCT, namely . However, they have not , because if one would, then from the "GCD LCM equation" that to associated needs to be. However, the common multiple is not a multiple of , so there is none and the two elements have none at all . ${\ displaystyle 1 + {\ sqrt {-3}}}$${\ displaystyle 2}$${\ displaystyle 1}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle v}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle v}$${\ displaystyle k: = (1 + {\ sqrt {-3}}) \ cdot 2}$${\ displaystyle 4}$${\ displaystyle k}$${\ displaystyle k}$${\ displaystyle \ mathrm {kgV}}$${\ displaystyle \ mathrm {kgV}}$

### Remarks

An integrity ring in which two elements have one is called a GCT ring or GCT area . In a GCD ring, every two elements also have one . ${\ displaystyle \ mathrm {ggT}}$${\ displaystyle \ mathrm {kgV}}$

In a factorial ring , every two elements have one . ${\ displaystyle \ mathrm {ggT}}$

In a Euclidean ring , the two elements can be determined with the Euclidean algorithm. ${\ displaystyle \ mathrm {ggT}}$