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least common multiple
In arithmetic and number theory, the least common multiple (also called the 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''.〔Hardy & Wright, § 5.1, p. 48〕 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 familiar from grade-school arithmetic as the "lowest common denominator" (LCD) that must be determined 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.
== 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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