least common multiple

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least common multiple

A student calculates the least common multiple of 12 and 18 on a whiteboard.

Definition

Noun: The smallest positive integer that is a multiple of each number in a given set. It is the smallest number into which all the numbers in the set can divide evenly without leaving a remainder.

Usage

The term "least common multiple" is used primarily in mathematics, specifically in arithmetic and number theory. It is a fundamental concept for operations with fractions, such as addition and subtraction, where finding a common denominator is necessary. - It is often abbreviated as LCM. - It is calculated for two or more integers.

Examples
  • To add the fractions 1/4 and 1/6, you must find the least common multiple of 4 and 6, which is 12.
  • The least common multiple of 3, 5, and 10 is 30.
  • In the equation, we simplified the expression by using the least common multiple of the denominators.
Advanced Usage
  • "To find the least common multiple": This is the standard phrase for the process of calculation, which can involve listing multiples, using prime factorization, or applying the relationship between LCM and GCD (Greatest Common Divisor).
  • The least common multiple is closely related to the greatest common divisor (GCD). For any two numbers, and , the product of the LCM and the GCD is equal to the product of and (i.e., LCM(a, b) × GCD(a, b) = a × b).
Variants and Related Words
  • LCM: The standard abbreviation.
  • Lowest Common Multiple: A synonym, more commonly used in British English.
  • Common Multiple: Any number that is a multiple of each number in a set. The LCM is the smallest of these.
  • Greatest Common Divisor (GCD) / Highest Common Factor (HCF): A related concept referring to the largest number that divides each number in a set without a remainder.
Synonyms
  • Smallest Common Multiple
  • Lowest Common Multiple (chiefly British)
Antonyms

There is no direct antonym. The conceptual opposite in terms of divisibility is the Greatest Common Divisor (GCD), which finds the largest common factor, not the smallest common multiple.

least common multiple

A student calculates the least common multiple of 12 and 18 on a whiteboard.

Noun
  1. the smallest multiple that is exactly divisible by every member of a set of numbers
    • the least common multiple of 12 and 18 is 36

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