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The Love Song of J. Alfred Prufrock
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The Love Song of J. Alfred Prufrock
by T. S. Eliot
First published in
June 1915 issue of Poetry[2]
Country
United States
Language
English
Publisher
magazine (1915): Harriet Monroe
chapbook (1917): The Egoist, Ltd. (London)[1]
Lines
140
Pages
6 (1915 printing)[2]
8 (1917 printing)[1]
Read online
The Love Song of J. Alfred Prufrock at Wikisource
"The Love Song of J. Alfred Prufrock", commonly known as "Prufrock", is the first professionally published poem by American-born, British poet T. S. Eliot (1888–1965). Eliot began writing "Prufrock" in February 1910, and it was first published in the June 1915 issue of Poetry: A Magazine of Verse[2] at the instigation of Ezra Pound (1885–1972). It was later printed as part of a twelve-poem pamphlet (or chapbook) titled Prufrock and Other Observations in 1917.[1] At the time of its publication, Prufrock was considered outlandish,[3] but is now seen as heralding a paradigmatic cultural shift from late 19th-century Romantic verse and Georgian lyrics to Modernism.
The poem's structure was heavily influenced by Eliot's extensive reading of Dante Alighieri[4] and makes several references to the Bible and other literary works—including William Shakespeare's plays Henry IV Part II, Twelfth Night, and Hamlet, the poetry of seventeenth-century metaphysical poet Andrew Marvell, and the nineteenth-century French Symbolists. Eliot narrates the experience of Prufrock using the stream of consciousness technique developed by his fellow Modernist writers. The poem, described as a "drama of literary anguish", is a dramatic interior monologue of an urban man, stricken with feelings of isolation and an incapability for decisive action that is said "to epitomize frustration and impotence of the modern individual" and "represent thwarted desires and modern disillusionment".[5]
Prufrock laments his physical and intellectual inertia, the lost opportunities in his life and lack of spiritual progress, and he is haunted by reminders of unattained carnal love. With visceral feelings of weariness, regret, embarrassment, longing, emasculation, sexual frustration, a sense of decay, and an awareness of mortality, "Prufrock" has become one of the most recognised voices in modern literature.[6]
Contents
[hide]
1 Composition and publication history
1.1 Writing and first publication
1.2 Prufrock's Pervigilium
1.3 Critical reception
2 Description
2.1 Title
2.2 Epigraph
2.3 Themes and interpretation
2.4 Use of allusion
3 Notes
4 Further reading
5 External links
Composition and publication history[edit]
T. S. Eliot in 1923, photographed by Lady Ottoline Morrell
Writing and first publication[edit]
Eliot wrote "The Love Song of J. Alfred Prufrock" between February 1910 and July or August 1911. Shortly after arriving in England to attend Merton College, Oxford, Eliot was introduced to American expatriate poet Ezra Pound, who instantly deemed Eliot "worth watching" and aided the start of Eliot's career. Pound served as the overseas editor of Poetry: A Magazine of Verse and recommended to the magazine's founder, Harriet Monroe, that Poetry publish "The Love Song of J. Alfred Prufrock", extolling that Eliot and his work embodied a new and unique phenomenon among contemporary writers. Pound claimed that Eliot "has actually trained himself AND modernized himself ON HIS OWN. The rest of the promising young have done one or the other, but never both."[7] The poem was first published by the magazine in its June 1915 issue.[2][8]
In November 1915 "The Love Song of J. Alfred Prufrock"—along with Eliot's poems "Portrait of a Lady," "The Boston Evening Transcript," "Hysteria," and "Miss Helen Slingsby"—was included in Catholic Anthology 1914–1915 edited by Ezra Pound and printed by Elkin Mathews in London.[9]:297 In June 1917 The Egoist, a small publishing firm run by Dora Marsden, published a pamphlet entitled Prufrock and Other Observations (London), containing twelve poems by Eliot. "The Love Song of J. Alfred Prufrock" was the first in the volume.[1] Also Eliot was appointed assistant editor of the Egoist in June 1917.[9]:290
Prufrock's Pervigilium[edit]
According to Eliot biographer Lyndall Gordon, when Eliot was writing the first drafts of Prufrock in his notebook in 1910–1911, he intentionally kept four pages blank in the middle section of the poem.[10] According to the notebooks, now in the collection of the New York Public Library, Eliot finished the poem that was originally published sometime in July and August 1911—when he was 22 years old.[11] In 1912, Eliot revised the poem and included a 38-line section now called "Prufrock's Pervigilium" which was inserted on those blank pages, and intended as a middle section for the poem.[10] However, Eliot removed this section soon after seeking the advice of his fellow Harvard acquaintance and poet Conrad Aiken.[12] This section would not be included in the original publication of Eliot's poem but was included when published posthumously in the 1996 collection of Eliot's early, unpublished drafts in Inventions of the March Hare: Poems 1909–1917.[11] This Pervigilium section describes the "vigil" of Prufrock through an evening and night[11]:41, 43–44, 176–90 described by one reviewer as an "erotic foray into the narrow streets of a social and emotional underworld" that portray "in clammy detail Prufrock's tramping 'through certain half-deserted streets' and the context of his 'muttering retreats / Of restless nights in one-night cheap hotels.'"[13]
Critical reception[edit]
Its reception in London can be gauged from an unsigned review in The Times Literary Supplement on 21 June 1917. "The fact that these things occurred to the mind of Mr. Eliot is surely of the very smallest importance to anyone, even to himself. They certainly have no relation to poetry."[14]
The Harvard Vocarium at Harvard College recorded Eliot's reading of Prufrock and other poems in 1947, as part of their ongoing series of poetry readings by their authors.[15]
Description[edit]
Title[edit]
In his early drafts, Eliot gave the poem the subtitle "Prufrock among the Women."[11]:41 This subtitle was apparently discarded before publication. Eliot called the poem a "love song" in reference to Rudyard Kipling's poem "The Love Song of Har Dyal," first published in Kipling's collection Plain Tales from the Hills (1888).[16] In 1959, Eliot addressed a meeting of the Kipling Society and discussed the influence of Kipling upon his own poetry:
Traces of Kipling appear in my own mature verse where no diligent scholarly sleuth has yet observed them, but which I am myself prepared to disclose. I once wrote a poem called "The Love Song of J. Alfred Prufrock": I am convinced that it would never have been called "Love Song" but for a title of Kipling's that stuck obstinately in my head: "The Love Song of Har Dyal".[16]
However, the origin of the name Prufrock is not certain, and Eliot never remarked on its origin other than to claim he was unsure of how he came upon the name. Many scholars and indeed Eliot himself have pointed towards the autobiographical elements in the character of Prufrock, and Eliot at the time of writing the poem was in the habit of rendering his name as "T. Stearns Eliot," very similar in form to that of J. Alfred Prufrock.[17] It is suggested that the name "Prufrock" came from Eliot's youth in St. Louis, Missouri, where the Prufrock-Litton Company, a large furniture store, occupied one city block downtown at 420–422 North Fourth Street.[18][19][20] In a 1950 letter, Eliot said, "I did not have, at the time of writing the poem, and have not yet recovered, any recollection of having acquired this name in any way, but I think that it must be assumed that I did, and that the memory has been obliterated."[21]
Epigraph[edit]
The draft version of the poem's epigraph comes from Dante's Purgatorio (XXVI, 147–148):[11]:39, 41
'sovegna vos a temps de ma dolor'.
Poi s'ascose nel foco che gli affina.
'be mindful in due time of my pain'.
Then dived he back into that fire which refines them.[22]
He finally decided not to use this, but eventually used the quotation in the closing lines of his 1922 poem The Waste Land. The quotation that Eliot did choose comes from Dante also. Inferno (XXVII, 61–66) reads:
S`io credesse che mia risposta fosse
A persona che mai tornasse al mondo,
Questa fiamma staria senza piu scosse.
Ma perciocchè giammai di questo fondo
Non tornò vivo alcun, s'i'odo il vero,
Senza tema d'infamia ti rispondo.
If I but thought that my response were made
to one perhaps returning to the world,
this tongue of flame would cease to flicker.
But since, up from these depths, no one has yet
returned alive, if what I hear is true,
I answer without fear of being shamed.[23]
In context, the epigraph refers to a meeting between Dante and Guido da Montefeltro, who was condemned to the eighth circle of Hell for providing counsel to Pope Boniface VIII, who wished to use Guido's advice for a nefarious undertaking. This encounter follows Dante's meeting with Ulysses, who himself is also condemned to the circle of the Fraudulent. According to Ron Banerjee, the epigraph serves to cast ironic light on Prufrock's intent. Like Guido, Prufrock had never intended his story to be told, and so by quoting Guido, Eliot reveals his view of Prufrock's love song.[24]
Frederick Locke contends that Prufrock himself is suffering from multiple personalities of sorts, and that he embodies both Guido and Dante in the Inferno analogy. One is the storyteller; the other the listener who later reveals the story to the world. He posits, alternatively, that the role of Guido in the analogy is indeed filled by Prufrock, but that the role of Dante is filled by you, the reader, as in "Let us go then, you and I," (1). In that, the reader is granted the power to do as he pleases with Prufrock's love song.[25]
Themes and interpretation[edit]
Because the poem is concerned primarily with the irregular musings of the narrator, it can be difficult to interpret. Laurence Perrine wrote, "[the poem] presents the apparently random thoughts going through a person's head within a certain time interval, in which the transitional links are psychological rather than logical".[26] This stylistic choice makes it difficult to determine exactly what is literal and what is symbolic. On the surface, "The Love Song of J. Alfred Prufrock" relays the thoughts of a sexually frustrated middle-aged man who wants to say something but is afraid to do so, and ultimately does not.[26][27] The dispute, however, lies in to whom Prufrock is speaking, whether he is actually going anywhere, what he wants to say, and to what the various images refer.
The intended audience is not evident. Some believe that Prufrock is talking to another person[28] or directly to the reader,[29] while others believe Prufrock's monologue is internal. Perrine writes "The 'you and I' of the first line are divided parts of Prufrock's own nature",[26] while Mutlu Konuk Blasing suggests that the "you and I" refers to the relationship between the dilemmas of the character and the author.[30] Similarly, critics dispute whether Prufrock is going somewhere during the course of the poem. In the first half of the poem, Prufrock uses various outdoor images (the sky, streets, cheap restaurants and hotels, fog), and talks about how there will be time for various things before "the taking of a toast and tea", and "time to turn back and descend the stair." This has led many to believe that Prufrock is on his way to an afternoon tea, where he is preparing to ask this "overwhelming question".[26] Others, however, believe that Prufrock is not physically going anywhere, but rather, is playing through it in his mind.[29][30]
Perhaps the most significant dispute lies over the "overwhelming question" that Prufrock is trying to ask. Many believe that Prufrock is trying to tell a woman of his romantic interest in her,[26] pointing to the various images of women's arms and clothing and the final few lines in which Prufrock laments that the mermaids will not sing to him. Others, however, believe that Prufrock is trying to express some deeper philosophical insight or disillusionment with society, but fears rejection, pointing to statements that express a disillusionment with society, such as "I have measured out my life with coffee spoons" (line 51). Many believe that the poem is a criticism of Edwardian society and Prufrock's dilemma represents the inability to live a meaningful existence in the modern world.[31] McCoy and Harlan wrote "For many readers in the 1920s, Prufrock seemed to epitomize the frustration and impotence of the modern individual. He seemed to represent thwarted desires and modern disillusionment."[29]
In general, Eliot uses imagery which is indicative of Prufrock's character,[26] representing ageing and decay. For example, "When the evening is spread out against the sky / Like a patient etherized upon a table" (lines 2–3), the "sawdust restaurants" and "cheap hotels," the yellow fog, and the afternoon "Asleep...tired... or it malingers" (line 77), are reminiscent of languor and decay, while Prufrock's various concerns about his hair and teeth, as well as the mermaids "Combing the white hair of the waves blown back / When the wind blows the water white and black," show his concern over aging.
Use of allusion[edit]
Like many of Eliot's poems, "The Love Song of J. Alfred Prufrock" makes numerous allusions to other works, which are often symbolic themselves.
In "Time for all the works and days of hands" (29) the phrase 'works and days' is the title of a long poem – a description of agricultural life and a call to toil – by the early Greek poet Hesiod.[26]
"I know the voices dying with a dying fall" (52) echoes Orsino's first lines in Shakespeare's Twelfth Night.[26]
The prophet of "Though I have seen my head (grown slightly bald) brought in upon a platter / I am no prophet — and here's no great matter" (81–2) is John the Baptist, whose head was delivered to Salome by Herod as a reward for her dancing (Matthew 14:1–11, and Oscar Wilde's play Salome).[26]
"To have squeezed the universe into a ball" (92) and "indeed there will be time" (23) echo the closing lines of Marvell's 'To His Coy Mistress'. Other phrases such as, "there will be time" and "there is time" are reminiscent of the opening line of that poem: "Had we but world enough and time". Marvell's words in turn echo the General Prologue of Chaucer's Canterbury Tales, "whil I have tyme and space".[26]
"'I am Lazarus, come from the dead'" (94) may be either the beggar Lazarus (of Luke 16) returning for the rich man who was not permitted to return from the dead to warn the brothers of a rich man about Hell, or the Lazarus (of John 11) whom Christ raised from the dead, or both.[26]
"Full of high sentence" (117) echoes Chaucer's description of the Clerk of Oxford in the General Prologue to The Canterbury Tales.[26]
"There will be time to murder and create" is a biblical allusion to Ecclesiastes 3.[26]
In the final section of the poem, Prufrock rejects the idea that he is Prince Hamlet, suggesting that he is merely "an attendant lord" (112) whose purpose is to "advise the prince" (114), a likely allusion to Polonius — Polonius being also "almost, at times, the Fool."
"Among some talk of you and me" may be[32] a reference to Quatrain 32 of Edward FitzGerald's translation of the Rubaiyat of Omar Khayyam ("There was a Door to which I found no Key / There was a Veil past which I could not see / Some little Talk awhile of Me and Thee / There seemed — and then no more of Thee and Me.")
Notes[edit]
^ Jump up to: a b c d Eliot, T. S. Prufrock and Other Observations (London: The Egoist, Ltd., 1917), 9–16.
^ Jump up to: a b c d Eliot, T. S. "The Love Song of J. Alfred Prufrock" in Monroe, Harriet (editor), Poetry: A Magazine of Verse (June 1915), 130–135.
Jump up ^ Eliot, T. S. (21 December 2010). The Waste Land and Other Poems. Broadview Press. p. 133. ISBN 978-1-77048-267-8. Retrieved 9 July 2017. (citing an unsigned review in Literary Review. 5 July 1917, vol. lxxxiii, 107.)
Jump up ^ Hollahan, Eugene (March 1970). "A Structural Dantean Parallel in Eliot's 'The Love Song of J. Alfred Prufrock'". American Literature. 1. 42: 91–93. doi:10.2307/2924384. ISSN 0002-9831.
Jump up ^ McCoy, Kathleen, and Harlan, Judith. English Literature From 1785 (New York: HarperCollins, 1992), 265–66. ISBN 006467150X
Jump up ^ Bercovitch, Sacvan. The Cambridge History of American Literature. Volume 5. (Cambridge: Cambridge University Press, 2003), 99. ISBN 0521497310
Jump up ^ Capitalization and italics original. Quoted in Mertens, Richard. "Letter By Letter" in The University of Chicago Magazine (August 2001). Retrieved 23 April 2007.
Jump up ^ Southam, B.C. A Guide to the Selected Poems of T.S. Eliot. (New York: Harcourt, Brace & Company, 1994), 45. ISBN 057117082X
^ Jump up to: a b Miller, James Edward. T.S. Eliot: The Making of an American poet, 1888–1922. (State College, Pennsylvania: Pennsylvania State University Press, 2005) ISBN 0271026812
^ Jump up to: a b Gordon, Lyndall. Eliot's New Life. (Oxford: Oxford University Press, 1988), 45.
^ Jump up to: a b c d e Eliot, T. S., and Ricks, Christopher B. (editor). Inventions of the March Hare: Poems 1909–1917 Ed. Christopher B. Ricks. (New York: Harcourt, 1996).
Jump up ^ Mayer, Nicholas B (2011). "Catalyzing Prufrock". Journal of Modern Literature. 34 (3): 182. doi:10.2979/jmodelite.34.3.182. JSTOR 10.2979/jmodelite.34.3.182.
Jump up ^ Jenkins, Nicholas. "More American Than We Knew: Nerves, exhaustion and madness were at the core of Eliot's early imaginative thinking" in The New York Times (20 April 1997). This is a 1997 book review of Inventions of the March Hare:Poems 1909–1917, vide supra. Retrieved 12 June 2013.
Jump up ^ Waugh, Arthur. The New Poetry, Quarterly Review, October 1916, citing the Times Literary Supplement 21 June 1917, no. 805, 299; Wagner, Erica (2001) "An eruption of fury", The Guardian, letters to the editor, 4 September 2001. Wagner omits the word "very" from the quote.
Jump up ^ Woodberry Poetry Room (Harvard College Library). Poetry Readings: Guide
^ Jump up to: a b Eliot, T. S. "The Unfading Genius of Rudyard Kipling" in Kipling Journal (March 1959), 9.
Jump up ^ Eliot, T. S. The Letters of T. S. Eliot. (New York: Harcourt, Brace Jovanovich, 1988). 1:135.
Jump up ^ Montesi, Al, and Deposki, Richard. Downtown St. Louis (Arcadia Publishing, 2001), 65. ISBN 0-7385-0816-0
Jump up ^ Christine H. The Daily Postcard: Prufrock-Litton – St. Louis, Missouri. Retrieved 21 February 2012.
Jump up ^ Missouri History Museum. Lighting fixture in front of Prufrock-Litton Furniture Company. Retrieved 11 June 2013.
Jump up ^ Stepanchev, Stephen. "The Origin of J. Alfred Prufrock" in Modern Language Notes. (1951), 66:400–401. JSTOR 2909497
Jump up ^ Eliot provided this translation in his essay "Dante" (1929).
Jump up ^ Dante Alighieri, and Hollander Robert and Hollander, Jean (translators), The Inferno. (Princeton: Princeton Dante Project). Retrieved 3 November 2011.
Jump up ^ Banerjee, Ron D. K. "The Dantean Overview: The Epigraph to 'Prufrock'" in Comparative Literature. (1972) 87:962–966. JSTOR 2907793
Jump up ^ Locke, Frederick W. "Dante and T. S. Eliot's Prufrock." in Modern Language Notes. (1963) 78:51–59. JSTOR 3042942
^ Jump up to: a b c d e f g h i j k l m Perrine, Laurence. Literature: Structure, Sound, and Sense, 1st edition. (New York: Harcourt, Brace & World, 1956), 798.
Jump up ^ On 'The Love Song of J. Alfred Prufrock' (accessed 14 June 2006).
Jump up ^ Headings, Philip R. T. S. Eliot. (Boston: Twayne Publishers, 1982), 24–25.
^ Jump up to: a b c Hecimovich, Gred A (editor). English 151-3; T. S. Eliot "The Love Song of J. Alfred Prufrock" notes (accessed 14 June 2006), from McCoy, Kathleen; Harlan, Judith. English Literature from 1785. (New York: HarperCollins, 1992).
^ Jump up to: a b Blasing, Mutlu Konuk, "On 'The Love Song of J. Alfred Prufrock'", in American Poetry: The Rhetoric of Its Forms (New Haven: Yale University Press, 1987). ISBN 0300037937
Jump up ^ Mitchell, Roger. "On 'The Love Song of J. Alfred Prufrock'", in Myers, Jack and Wojahan, David (editors). A Profile of Twentieth-Century American Poetry. (Carbondale, Illinois: Southern Illinois University Press, 1991). ISBN 0809313480
Jump up ^ Schimanski, Johan Annotasjoner til T. S. Eliot, "The Love Song of J. Alfred Prufock" (at Universitetet i Tromsø). Retrieved 8 August 2006.
Further reading[edit]
Drew, Elizabeth. T. S. Eliot: The Design of His Poetry (New York: Charles Scribner's Sons, 1949).
Gallup, Donald. T. S. Eliot: A Bibliography (A Revised and Extended Edition) (New York: Harcourt Brace & World, 1969), 23, 196.
Luthy, Melvin J. "The Case of Prufrock's Grammar" in College English (1978) 39:841–853. JSTOR 375710
Soles, Derek. "The Prufrock Makeover" in The English Journal (1999), 88:59–61. JSTOR 822420
Sorum, Eve. "Masochistic Modernisms: A Reading of Eliot and Woolf." Journal of Modern Literature. 28 (3), (Spring 2005) 25–43. doi:10.1353/jml.2005.0044
Sinha, Arun Kumar and Vikram, Kumar. "'The Love Song of J Alfred Prufrock' (Critical Essay with Detailed Annotations)" in T. S. Eliot: An Intensive Study of Selected Poems (New Delhi: Spectrum Books Pvt. Ltd, 2005).
Walcutt, Charles Child. "Eliot's "The Love Song of J. Alfred Prufrock" in College English (1957) 19:71–72. JSTOR 372706
External links[edit]
Wikisource has original text related to this article:
The Love Song of J. Alfred Prufrock
Original text from Poetry magazine June 1915
Text and extended audio discussion of the poem
The Love Song of J. Alfred Prufrock at the British Library
Prufrock and Other Observations at Project Gutenberg
Annotated hypertext version of the poem
Love Song of J. Alfred Prufrock public domain audiobook at LibriVox
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Least common multiple
From Wikipedia, the free encyclopedia
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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.[1] Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.[2] 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
[hide]
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[edit]
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[edit]
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].[2][3]
The programming language J uses a*.b
Example[edit]
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[edit]
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[edit]
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[edit]
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.[4]
Computing the least common multiple[edit]
Reduction by the greatest common divisor[edit]
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[edit]
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 23, 32, and 71, 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[edit]
This method works as easily for finding the LCM of several integers.
Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X, where X(m) 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(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as
xk(m+1) = xk(m), k ≠ k0
xk0(m+1) = xk0(m) + xk0(1).
In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged.
The algorithm stops when all elements in sequence X(m) are equal. Their common value L is exactly LCM(X).
A method using a table[edit]
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
2
4 2
7 7
12 6
21 21
42 21
Now, check if 2 divides again:
x
2
2
4 2 1
7 7 7
12 6 3
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
2
2
3
7
4 2 1 1 1
7 7 7 7 1
12 6 3 1 1
21 21 21 7 1
42 21 21 7 1
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[edit]
Fundamental theorem of arithmetic[edit]
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 n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...
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[edit]
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 isn'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[5] 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[edit]
Let D be the product of ω(D) distinct prime numbers (i.e. D is squarefree).
Then[6]
| { ( 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[edit]
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[edit]
Anomalous cancellation
Chebyshev function
Greatest common divisor
Notes[edit]
Jump up ^ Hardy & Wright, § 5.1, p. 48
^ Jump up to: a b 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[edit]
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=828245491"
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\operatorname{lcm}(21,6)
={21\cdot6\over\operatorname{gcd}(21,6)}
={21\cdot6\over\operatorname{gcd}(3,6)}
={21\cdot 6\over 3}= \frac{126}{3} = 42.

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values: {\displaystyle \gcd(a,b)\operatorname {lcm} (a,b)=a\,b.}

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Partially ordered

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Meet (mathematics)

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Commutative operation

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values: {\displaystyle \operatorname {lcm} (a,\gcd(a,b))=a,}

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values: {\displaystyle \gcd(a,\operatorname {lcm} (a,b))=a.}

values: {\displaystyle \operatorname {lcm} (a,a)=a,}

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values: {\displaystyle a\geq b\iff a=\operatorname {lcm} (a,b),}

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values: {\displaystyle a\leq b\iff a=\gcd(a,b).}

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values: Edit section: The LCM in commutative rings

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