Least common multiple

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In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. If there is no such positive integer, e.g., if a = 0 or b = 0, then lcm(a, b) is defined to be zero.

The least common multiple is useful when adding or subtracting vulgar fractions, because it yields the lowest common denominator. Consider for instance

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {2\over21}+{1\over6}={4\over42}+{7\over42}={11\over42},}

where the denominator 42 was used because lcm(21, 6) = 42.

If a and b are not both zero, the least common multiple can be computed by using the greatest common divisor (gcd) of a and b:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(a,b)=\frac{a\cdot b}{\operatorname{gcd}(a,b)}.}

Thus, the Euclidean algorithm for the gcd also gives us a fast algorithm for the lcm. To return to the example above,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(21,6) ={21\cdot6\over\operatorname{gcd}(21,6)} ={21\cdot 6\over 3}={21\cdot 2}=42.}

Efficient calculation

The formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(a,b)=\frac{(a\cdot b)}{\operatorname{gcd}(a,b)}}

is adequate to calculate the lcm for small numbers using the formula as written.

Because that (ab)/c = a(b/c) = (a/c)b, one can calculate the lcm using the above formula more efficiently, by firstly exploiting the fact that b/c or a/c may be easier to calculate than the quotient of the product ab and c. This can be true whether the calculations are performed by a human, or a computer, which may have storage requirements on the variables a, b, c, where the limits may be 4 byte storage - calculating ab may cause an overflow, if storage space is not allocated properly.

Using this, we can then calculate the lcm by either using:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(a,b)=\left({a\over\operatorname{gcd}(a,b)}\right)\cdot b}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(a,b)=a\cdot\left({b\over\operatorname{gcd}(a,b)}\right).\,}

Done this way, the previous example becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(21,6)={21\over\operatorname{gcd}(21,6)}\cdot6={21\over3}\cdot6=7\cdot6=42.}

Alternative method

The unique factorization theorem says that every positive integer number greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number.

For example:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 90 = 2^1 \cdot 3^2 \cdot 5^1 = 2 \cdot 9 \cdot 5 \,\!}

Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5.

We can use this knowledge to easily find the lcm of a group of numbers.

For example: Find the value of lcm(45, 120, 75)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 45\; \, = 2^0 \cdot 3^2 \cdot 5^1 \,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 120 = 2^3 \cdot 3^1 \cdot 5^1 \,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 75\; \,= 2^0 \cdot 3^1 \cdot 5^2. \,\!}

The lcm is the number which has the greatest multiple of each different type of atom. Thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{lcm}(45,120,75) = 2^3 \cdot 3^2 \cdot 5^2 = 8 \cdot 9 \cdot 25 = 1800. \,\!}

See also

External links

ca:Mínim comú múltiple de:Größter gemeinsamer Teiler und kleinstes gemeinsames Vielfaches es:Mínimo común múltiplo eo:Plej malgranda komuna oblo fr:Plus petit commun multiple ko:최소공배수 it:Minimo comune multiplo nl:Kleinste gemene veelvoud ja:最小公倍数 pl:Najmniejsza wspólna wielokrotność fi:Pienin yhteinen jaettava zh:最小公倍數