Least common multiple - wikipedia Least common multiple LCMs of numbers 1 through 10 with numbers 2 through 10. Line labels = first number. X axis = second number minus 1. Y axis = LCM of the two numbers. A Venn diagram showing the least common multiples of combinations of 2, 3, 4, 5 and 7 ( 6 is skipped as it is 2 × 3, both of which are already represented ). For example, a card game which requires its cards to be divided equally among up to 5 players requires at least 60 cards, the number at the intersection of the 2, 3, 4 and 5 sets, but not the 7 set. In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM ( a, b ), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm ( a, 0 ) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility. The LCM is the `` lowest common denominator '' ( LCD ) that can be used before fractions can be added, subtracted or compared. The LCM of more than two integers is also well - defined : it is the smallest positive integer that is divisible by each of them. Contents 1 Overview 1.1 Notation 1.2 Example 1.3 Applications 1.3. 1 Gears problem 1.3. 2 Planetary alignment 2 Computing the least common multiple 2.1 Reduction by the greatest common divisor 2.2 Finding least common multiples by prime factorization 2.3 A simple algorithm 2.4 A method using a table 3 Formulas 3.1 Fundamental theorem of arithmetic 3.2 Lattice - theoretic 3.3 Other 4 The LCM in commutative rings 5 See also 6 Notes 7 References Overview A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of − 5 and − 2 as well. Notation In this article we will denote the least common multiple of two integers a and b as lcm ( a, b ). Some older textbooks use ( a, b ). The programming language J uses a *. b Example What is the LCM of 4 and 6 ? Multiples of 4 are : 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76,... and the multiples of 6 are : 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72,... Common multiples of 4 and 6 are simply the numbers that are in both lists : 12, 24, 36, 48, 60, 72,... So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12. Applications When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, 2 21 + 1 6 = 4 42 + 7 42 = 11 42 ( displaystyle ( 2 over 21 ) + ( 1 over 6 ) = ( 4 over 42 ) + ( 7 over 42 ) = ( 11 over 42 ) ) where the denominator 42 was used because it is the least common multiple of 21 and 6. Gears problem Suppose there are two meshing gears in a machine, having m and n teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, we can determine how many rotations the first gear must complete to realign the line segment by making use of LCM ( m, n ). The first gear must complete LCM ( m, n ) / m rotations for the realignment. By that time, the second gear will have made LCM ( m, n ) / n rotations. Planetary alignment Suppose there are three planets revolving around a star which take l, m and n units of time respectively to complete their orbits. Assume that l, m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after LCM ( l, m, n ) units of time. At this time, the first, second and third planet will have completed LCM ( l, m, n ) / l, LCM ( l, m, n ) / m and LCM ( l, m, n ) / n orbits respectively around the star. Computing the least common multiple Reduction by the greatest common divisor The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor ( GCD ), also known as the greatest common factor : lcm ⁡ ( a, b ) = a ⋅ b gcd ⁡ ( a, b ). ( displaystyle operatorname ( lcm ) ( a, b ) = ( \frac ( a cdot b ) ( operatorname ( gcd ) ( a, b ) ) ). ) This formula is also valid when exactly one of a and b is 0, since gcd ( a, 0 ) = a. However, if both a and b are 0, this formula would cause division by zero ; lcm ( 0, 0 ) = 0 is a special case. There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, lcm ⁡ ( 21, 6 ) = 21 ⋅ 6 gcd ⁡ ( 21, 6 ) = 21 ⋅ 6 gcd ⁡ ( 3, 6 ) = 21 ⋅ 6 3 = 126 3 = 42. ( displaystyle operatorname ( lcm ) ( 21, 6 ) = ( 21 cdot 6 over operatorname ( gcd ) ( 21, 6 ) ) = ( 21 cdot 6 over operatorname ( gcd ) ( 3, 6 ) ) = ( 21 cdot 6 over 3 ) = ( \frac ( 126 ) ( 3 ) ) = 42. ) Because gcd ( a, b ) is a divisor of both a and b, it is more efficient to compute the LCM by dividing before multiplying : lcm ⁡ ( a, b ) = ( a gcd ⁡ ( a, b ) ) ⋅ b = ( b gcd ⁡ ( a, b ) ) ⋅ a. ( displaystyle operatorname ( lcm ) ( a, b ) = left ( ( a over operatorname ( gcd ) ( a, b ) ) \right ) cdot b = left ( ( b over operatorname ( gcd ) ( a, b ) ) \right ) cdot a. ) This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results ( overflow in the a × b computation ). Because gcd ( a, b ) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes : lcm ⁡ ( 21, 6 ) = 21 gcd ⁡ ( 21, 6 ) ⋅ 6 = 21 gcd ⁡ ( 3, 6 ) ⋅ 6 = 21 3 ⋅ 6 = 7 ⋅ 6 = 42. ( displaystyle operatorname ( lcm ) ( 21, 6 ) = ( 21 over operatorname ( gcd ) ( 21, 6 ) ) cdot 6 = ( 21 over operatorname ( gcd ) ( 3, 6 ) ) cdot 6 = ( 21 over 3 ) cdot 6 = 7 cdot 6 = 42. ) Finding least common multiples by prime factorization The unique factorization theorem says that every positive integer 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 : 90 = 2 1 ⋅ 3 2 ⋅ 5 1 = 2 ⋅ 3 ⋅ 3 ⋅ 5. ( displaystyle 90 = 2 ^ ( 1 ) cdot 3 ^ ( 2 ) cdot 5 ^ ( 1 ) = 2 cdot 3 cdot 3 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. This knowledge can be used to find the LCM of a set of numbers. Example : Find the value of lcm ( 8, 9, 21 ). First, factor each number and express it as a product of prime number powers. 8 = 2 3 9 = 3 2 21 = 3 1 ⋅ 7 1 ( displaystyle ( \begin ( aligned ) 8& = 2 ^ ( 3 ) 9& = 3 ^ ( 2 ) 21& = 3 ^ ( 1 ) cdot 7 ^ ( 1 ) end ( aligned ) ) ) The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 2, 3, and 7, respectively. Thus, lcm ⁡ ( 8, 9, 21 ) = 2 3 ⋅ 3 2 ⋅ 7 1 = 8 ⋅ 9 ⋅ 7 = 504. ( displaystyle operatorname ( lcm ) ( 8, 9, 21 ) = 2 ^ ( 3 ) cdot 3 ^ ( 2 ) cdot 7 ^ ( 1 ) = 8 cdot 9 cdot 7 = 504. ) This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts. This method can be illustrated using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram. Here is an example : 48 = 2 × 2 × 2 × 2 × 3, 180 = 2 × 2 × 3 × 3 × 5, and what they share in common is two `` 2 '' s and a `` 3 '' : Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720 Greatest common divisor = 2 × 2 × 3 = 12 This also works for the greatest common divisor ( GCD ), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the GCD of 48 and 180 is 2 × 2 × 3 = 12. A simple algorithm This method works as easily for finding the LCM of several integers. Let there be a finite sequence of positive integers X = ( x, x,..., x ), n > 1. The algorithm proceeds in steps as follows : on each step m it examines and updates the sequence X = ( x, x,..., x ), X = X, where X is the mth iteration of X, i.e. X at step m of the algorithm, etc. The purpose of the examination is to pick the least ( perhaps, one of many ) element of the sequence X. Assuming x is the selected element, the sequence X is defined as x = x, k ≠ k x = x + x. In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X to X unchanged. The algorithm stops when all elements in sequence X are equal. Their common value L is exactly LCM ( X ). A method using a table This method works for any number of factors. One begins by listing all of the numbers vertically in a table ( in this example 4, 7, 12, 21, and 42 ) : 4 7 12 21 42 The process begins by dividing all of the factors by 2. If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2. If a number does not divide evenly, just rewrite the number again. If 2 does not divide evenly into any of the numbers, try 3. x 7 7 12 6 21 21 42 21 Now, check if 2 divides again : x 7 7 7 12 6 21 21 21 42 21 21 Once 2 no longer divides, divide by 3. If 3 no longer divides, try 5 and 7. Keep going until all of the numbers have been reduced to 1. x 7 7 7 7 7 12 6 21 21 21 7 42 21 21 7 Now, multiply the numbers on the top and you have the LCM. In this case, it is 2 × 2 × 3 × 7 = 84. You will get to the LCM the quickest if you use prime numbers and start from the lowest prime, 2. As a general computational algorithm, the above is quite inefficient. One would never want to implement it in software : it takes too many steps, and requires too much storage space. A far more efficient numerical algorithm can be obtained simply by using Euclid 's algorithm to compute the gcd first, and then obtaining the lcm by division. Formulas Fundamental theorem of arithmetic According to the fundamental theorem of arithmetic a positive integer is the product of prime numbers, and, except for their order, this representation is unique : n = 2 n 2 3 n 3 5 n 5 7 n 7 ⋯ = ∏ p p n p, ( displaystyle n = 2 ^ ( n_ ( 2 ) ) 3 ^ ( n_ ( 3 ) ) 5 ^ ( n_ ( 5 ) ) 7 ^ ( n_ ( 7 ) ) cdots = prod _ ( p ) p ^ ( n_ ( p ) ), ) where the exponents n, n,... are non-negative integers ; for example, 84 = 2 3 5 7 11 13... Given two positive integers a = ∏ p p a p ( displaystyle ; a = prod _ ( p ) p ^ ( a_ ( p ) ) ) and b = ∏ p p b p ( displaystyle ; b = prod _ ( p ) p ^ ( b_ ( p ) ) ) their least common multiple and greatest common divisor are given by the formulas gcd ( a, b ) = ∏ p p min ( a p, b p ) ( displaystyle gcd ( a, b ) = prod _ ( p ) p ^ ( min ( a_ ( p ), b_ ( p ) ) ) ) and lcm ⁡ ( a, b ) = ∏ p p max ( a p, b p ). ( displaystyle operatorname ( lcm ) ( a, b ) = prod _ ( p ) p ^ ( max ( a_ ( p ), b_ ( p ) ) ). ) Since min ( x, y ) + max ( x, y ) = x + y, ( displaystyle min ( x, y ) + max ( x, y ) = x + y, ) this gives gcd ( a, b ) lcm ⁡ ( a, b ) = a b. ( displaystyle gcd ( a, b ) operatorname ( lcm ) ( a, b ) = a , b. ) In fact, any rational number can be written uniquely as the product of primes if negative exponents are allowed. When this is done, the above formulas remain valid. For example : 4 = 2 2 3 0, 6 = 2 1 3 1, gcd ( 4, 6 ) = 2 1 3 0, lcm ⁡ ( 4, 6 ) = 2 2 3 1. 1 3 = 2 0 3 − 1 5 0, 2 5 = 2 1 3 0 5 − 1, gcd ( 1 3, 2 5 ) = 2 0 3 − 1 5 − 1, lcm ⁡ ( 1 3, 2 5 ) = 2 1 3 0 5 0, 1 6 = 2 − 1 3 − 1, 3 4 = 2 − 2 3 1, gcd ( 1 6, 3 4 ) = 2 − 2 3 − 1, lcm ⁡ ( 1 6, 3 4 ) = 2 − 1 3 1. ( displaystyle ( \begin ( array ) ( llll ) 4 = 2 ^ ( 2 ) 3 ^ ( 0 ), &6 = 2 ^ ( 1 ) 3 ^ ( 1 ), & gcd ( 4, 6 ) = 2 ^ ( 1 ) 3 ^ ( 0 ), & operatorname ( lcm ) ( 4, 6 ) = 2 ^ ( 2 ) 3 ^ ( 1 ). ( 8pt ) ( \tfrac ( 1 ) ( 3 ) ) = 2 ^ ( 0 ) 3 ^ ( - 1 ) 5 ^ ( 0 ), & ( \tfrac ( 2 ) ( 5 ) ) = 2 ^ ( 1 ) 3 ^ ( 0 ) 5 ^ ( - 1 ), & gcd ( ( \tfrac ( 1 ) ( 3 ) ), ( \tfrac ( 2 ) ( 5 ) ) ) = 2 ^ ( 0 ) 3 ^ ( - 1 ) 5 ^ ( - 1 ), & operatorname ( lcm ) ( ( \tfrac ( 1 ) ( 3 ) ), ( \tfrac ( 2 ) ( 5 ) ) ) = 2 ^ ( 1 ) 3 ^ ( 0 ) 5 ^ ( 0 ), ( 8pt ) ( \tfrac ( 1 ) ( 6 ) ) = 2 ^ ( - 1 ) 3 ^ ( - 1 ), & ( \tfrac ( 3 ) ( 4 ) ) = 2 ^ ( - 2 ) 3 ^ ( 1 ), & gcd ( ( \tfrac ( 1 ) ( 6 ) ), ( \tfrac ( 3 ) ( 4 ) ) ) = 2 ^ ( - 2 ) 3 ^ ( - 1 ), & operatorname ( lcm ) ( ( \tfrac ( 1 ) ( 6 ) ), ( \tfrac ( 3 ) ( 4 ) ) ) = 2 ^ ( - 1 ) 3 ^ ( 1 ). end ( array ) ) ) Lattice - theoretic The positive integers may be partially ordered by divisibility : if a divides b ( i.e. if b is an integer multiple of a ) write a ≤ b ( or equivalently, b ≥ a ). ( Forget the usual magnitude - based definition of ≤ in this section - it is n't used. ) Under this ordering, the positive integers become a lattice with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious ; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them : If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. ( Remember ≤ is defined as divides ). The following pairs of dual formulas are special cases of general lattice - theoretic identities. Commutative laws lcm ⁡ ( a, b ) = lcm ⁡ ( b, a ), ( displaystyle operatorname ( lcm ) ( a, b ) = operatorname ( lcm ) ( b, a ), ) gcd ( a, b ) = gcd ( b, a ). ( displaystyle gcd ( a, b ) = gcd ( b, a ). ) Associative laws lcm ⁡ ( a, lcm ⁡ ( b, c ) ) = lcm ⁡ ( lcm ⁡ ( a, b ), c ), ( displaystyle operatorname ( lcm ) ( a, operatorname ( lcm ) ( b, c ) ) = operatorname ( lcm ) ( operatorname ( lcm ) ( a, b ), c ), ) gcd ( a, gcd ( b, c ) ) = gcd ( gcd ( a, b ), c ). ( displaystyle gcd ( a, gcd ( b, c ) ) = gcd ( gcd ( a, b ), c ). ) Absorption laws lcm ⁡ ( a, gcd ( a, b ) ) = a, ( displaystyle operatorname ( lcm ) ( a, gcd ( a, b ) ) = a, ) gcd ( a, lcm ⁡ ( a, b ) ) = a. ( displaystyle gcd ( a, operatorname ( lcm ) ( a, b ) ) = a. ) Idempotent laws lcm ⁡ ( a, a ) = a, ( displaystyle operatorname ( lcm ) ( a, a ) = a, ) gcd ( a, a ) = a. ( displaystyle gcd ( a, a ) = a. ). Define divides in terms of lcm and gcd a ≥ b ⟺ a = lcm ⁡ ( a, b ), ( displaystyle a geq b iff a = operatorname ( lcm ) ( a, b ), ) a ≤ b ⟺ a = gcd ( a, b ). ( displaystyle a leq b iff a = gcd ( a, b ). ) It can also be shown that this lattice is distributive, i.e. that lcm distributes over gcd and, dually, that gcd distributes over lcm : lcm ⁡ ( a, gcd ( b, c ) ) = gcd ( lcm ⁡ ( a, b ), lcm ⁡ ( a, c ) ), ( displaystyle operatorname ( lcm ) ( a, gcd ( b, c ) ) = gcd ( operatorname ( lcm ) ( a, b ), operatorname ( lcm ) ( a, c ) ), ) gcd ( a, lcm ⁡ ( b, c ) ) = lcm ⁡ ( gcd ( a, b ), gcd ( a, c ) ). ( displaystyle gcd ( a, operatorname ( lcm ) ( b, c ) ) = operatorname ( lcm ) ( gcd ( a, b ), gcd ( a, c ) ). ) This identity is self - dual : gcd ( lcm ⁡ ( a, b ), lcm ⁡ ( b, c ), lcm ⁡ ( a, c ) ) = lcm ⁡ ( gcd ( a, b ), gcd ( b, c ), gcd ( a, c ) ). ( displaystyle gcd ( operatorname ( lcm ) ( a, b ), operatorname ( lcm ) ( b, c ), operatorname ( lcm ) ( a, c ) ) = operatorname ( lcm ) ( gcd ( a, b ), gcd ( b, c ), gcd ( a, c ) ). ) Other Let D be the product of ω ( D ) distinct prime numbers ( i.e. D is squarefree ). Then ( ( x, y ) : lcm ⁡ ( x, y ) = D ) = 3 ω ( D ), ( displaystyle ( ( x, y ) ; : ; operatorname ( lcm ) ( x, y ) = D ) = 3 ^ ( omega ( D ) ), ) where the absolute bars denote the cardinality of a set. The lcm in commutative rings The least common multiple can be defined generally over commutative rings as follows : Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m ( i.e. there exist elements x and y of R such that ax = m and by = m ). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n. In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b ( the intersection of a collection of ideals is always an ideal ). See also Anomalous cancellation Chebyshev function Greatest common divisor Notes Jump up ^ Hardy & Wright, § 5.1, p. 48 ^ Jump up to : Long ( 1972, p. 39 ) Jump up ^ Pettofrezzo & Byrkit ( 1970, p. 56 ) Jump up ^ `` nasa spacemath '' ( PDF ). Jump up ^ The next three formulas are from Landau, Ex. III. 3, p. 254 Jump up ^ Crandall & Pomerance, ex. 2.4, p. 101. References Crandall, Richard ; Pomerance, Carl ( 2001 ), Prime Numbers : A Computational Perspective, New York : Springer, ISBN 0 - 387 - 94777 - 9 Hardy, G.H. ; Wright, E.M. ( 1979 ), An Introduction to the Theory of Numbers ( Fifth edition ), Oxford : Oxford University Press, ISBN 978 - 0 - 19 - 853171 - 5 Landau, Edmund ( 1966 ), Elementary Number Theory, New York : Chelsea Long, Calvin T. ( 1972 ), Elementary Introduction to Number Theory ( 2nd ed. ), Lexington : D.C. Heath and Company, LCCN 77 - 171950 Pettofrezzo, Anthony J. ; Byrkit, Donald R. ( 1970 ), Elements of Number Theory, Englewood Cliffs : Prentice Hall, LCCN 77 - 81766 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Least_common_multiple&oldid=812154127 '' Categories : Elementary arithmetic Arithmetic Talk Contents About Wikipedia Wikibooks Azərbaycanca বাংলা Bân - lâm - gú Беларуская Български Català Čeština Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Galego 한국어 Հայերեն हिन्दी Bahasa Indonesia Italiano עברית Қазақша Latviešu Lietuvių Magyar മലയാളം मराठी Bahasa Melayu Монгол Nederlands 日本 語 Norsk Norsk nynorsk پنجابی Piemontèis Plattdüütsch Polski Português Română Русский Shqip Simple English Slovenčina Slovenščina کوردی Српски / srpski Suomi Svenska Tagalog தமிழ் తెలుగు ไทย Türkçe Українська اردو Tiếng Việt ייִדיש 粵語 中文 Edit links This page was last edited on 26 November 2017, at 08 : 09. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. 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Question: what is the meaning of least common multiple
Answer: smallest positive integer that is divisible by both a and b