least common multiple

English

Noun

least common multiple (plural least common multiples)

1. (number theory) The smallest positive integer which is divisible by (equivalently, is an integer multiple of) each of a specified finite set of integers.

   120 is the least common multiple of 60, 8 and 15.

   • 1859, David Henry Cruttenden, The Objective Or Synthetic Arithmetic, J. M. Bradstreet & Son, page 261:

     The LEAST COMMON MULTIPLE is the smallest number, which can be measured by each of two, or more divisors.
     […]
     Since 0 is the least multiple of every number, it must be the least common multiple of any, or of all numbers. It is, however, excluded by common consent in finding the least common multiples of numbers, because as such, 0 has no practical value.

   • 2000, Robert W. Smith, Math Challenges, Grades 4-6, Teacher Created Resources, page 75:

     The least common multiple (LCM) is the smallest common multiple of two or more factors. […] The least common multiple is used to determine the appropriate denominator for adding unlike fractions. In adding unlike fractions, it is important to find the lowest common denominator of the fractions. This is exactly the same thing as finding the least common multiple of the denominators.

   • 2007, Jerome Kaufmann, Karen Schwitters, Algebra for College Students, 8th edition, Thomson Learning, page 820:

     It is sometimes necessary to determine the smallest common nonzero multiple of two or more whole numbers. We call this nonzero multiple the least common multiple. In our work with fractions, there will be problems where it will be necessary to find the least common multiple of some numbers, usually the denominators of fractions.

Usage notes

• The requirement that the least common multiple be positive has two effects:
  • It excludes zero, which is trivially divisible by any integer.
  • It means that the specified set cannot contain an element equal to zero. (This is also achieved by the requirement that the numbers be divisors, zero not being a valid divisor of any number.)
• Notations used include $\operatorname {LCM} (a,b),\ \operatorname {lcm} (a,b),\ \operatorname {l.c.m.} (a,b)$ and $\ [a,b]$.

Synonyms

• ((number theory)): lowest common multiple, smallest common multiple, lcm (initialism)

Coordinate terms

• greatest common divisor

Related terms

• least common denominator
• lowest common denominator

Translations

(number theory)

• Chinese:

  Mandarin: 最小公倍數／最小公倍数 (zh) (zuìxiǎo gōngbèishù)

• Danish: mindste fælles multiplum
• Dutch: kleinste gemene veelvoud
• Estonian: vähim ühiskordne
• Finnish: pienin yhteinen jaettava, pienin yhteinen monikerta
• French: plus petit commun multiple (fr) m
• Georgian: უმცირესი საერთო ჯერადი (umciresi saerto ǯeradi), საერთო უმცირესი ჯერადი (saerto umciresi ǯeradi)
• German: kleinstes gemeinsames Vielfaches n
• Greek: ελάχιστο κοινό πολλαπλάσιο (el) n (eláchisto koinó pollaplásio)
• Hungarian: legkisebb közös többszörös (hu)
• Indonesian: kelipatan persekutuan terkecil (id)
• Italian: minimo comune multiplo (it) m
• Japanese: 最小公倍数 (ja) (saishō kōbaisū)
• Korean: 최소공배수 (choesogongbaesu)
• Malay: gandaan sepunya terkecil
• Polish: najmniejsza wspólna wielokrotność f
• Romanian: cel mai mic mutiplu comun m, c.m.m.m.c.
• Russian: наиме́ньшее о́бщее кра́тное n (naiménʹšeje óbščeje krátnoje), НОК (ru) n (NOK)
• Spanish: mínimo común múltiplo m
• Tagalog: pinakamaliit na lahatang kaparami
• Turkish: ortak katların en küçüğü
• Ukrainian: найме́нше спі́льне кра́тне n (najménše spílʹne krátne), НСК n (NSK) (abbreviation)
• Vietnamese: bội chung nhỏ nhất, bội số chung nhỏ nhất

References

• 2008, D. R. Heath-Brown, J. H. Silverman (6th ed. editors), G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition Oxford University Press, Paperback, page 57.

Further reading

• Greatest common divisor on Wikipedia.Wikipedia
• Least common multiple on Encyclopedia of Mathematics
• Least Common Multiple on Wolfram MathWorld