If and are two positive integers, then their least common multiple, denoted by
is the positive integer satisfying the conditions
if and , then .
Note: The definition can be generalized for several numbers. The positive lcm of positive integers is uniquely determined. (Its negative satisfies the same two conditions.)
If and are the prime factor presentations of the positive integers and (, ), then
This can be generalized for lcm of several numbers.
Because the greatest common divisor has the expression , we see that
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,
The preceding formula may be presented in terms of ideals of ; we may replace the integers with the corresponding principal ideals. The formula acquires the form
The recent formula is valid also for other than principal ideals and even in so general systems as the Prüfer rings; in fact, it could be taken as defining property of these rings: Let be a commutative ring with non-zero unity. is a Prüfer ring iff Jensen’s formula
is true for all ideals and of , with at least one of them having non-zero-divisors.
- 1 M. Larsen and P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).