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. If either a or b is 0, LCM(a, b) is defined to be zero.
The LCM is familiar from grade-school arithmetic as the "least 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 integer that is divisible by each of them. multiple of a
number is the product of that number and an integer. For example, 10 is a multiple of 5 because , 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.In this article we will denote the least common multiple of two integers a and b as lcm( a, b ).
Some older textbooks use [ a, b ], ExampleWhat is the LCM of 4 and 6?
Multiples of 4 are:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, ...Know More About :- Binary Number System Math.Edurite.comPage : 1/3
and the multiples of 6 are:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
Common multiples of 4 and 6 are simply the numbers that are in both lists:
12, 24, 36, 48, 60, 72, ....
So the least common multiple of 4 and 6 is the smallest one of those: 12The LCM in commutative rings :-
The least common multiple can be defined generally over
commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of
a and b is an element m of R such that both a and b divide m (i.e. there exist elements x and y of R such
that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal in
the sense that for any other common multiple n of a and b, m divides n. In general, two elements in a
commutative ring can have no least common multiple or more than one. However, any two least
common multiples of the same pair of elements are associates. In a unique factorization domain, any
two elements have a least common multiple. In a principal ideal domain, the least common multiple of
a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the
intersection of a collection of ideals is always an ideal). In principal ideal domains, one can even talk
about the least common multiple of arbitrary collections of elements: it is a generator of the intersection
of the ideals generated by the elements of the collection.Read More About :- Composite Numbers Math.Edurite.comPage : 2/3