Least common multiple

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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 a multiple of both a and b.[1] It is familiar from grade-school arithmetic as the "lowest common denominator" that must be determined before two fractions can be added.

This definition may be extended to rational numbers a and b: the LCM is the smallest positive rational number that is an integer multiple of both a and b. (In fact, the definition may be extended to any two real numbers whose ratio is a rational number.)

If either a or b is 0, LCM(ab) is defined to be zero.

The LCM of more than two integers or rational numbers is well-defined: it is the smallest number that is an integer multiple of each of them.

Contents

Examples

Integer

What is the LCM of 4 and 6?

Multiples of 4 are:

and the multiples of 6 are:

Common multiples of 4 and 6 are simply the numbers that are in both lists:

So the least common multiple of 4 and 6 is the smallest one of those: 12 = 3 × 4 = 2 × 6.

Rational

What is the LCM of \tfrac{1}{3} and \tfrac{2}{5}?

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