In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If there is no such positive integer, e.g., if a = 0 or b = 0, then lcm(a, b) is defined to be zero. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The integers are commonly denoted by the above symbol. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount left over— is also acknowledged. ...

For example, the least common multiple of the numbers 4 and 6 is 12.

When adding or subtracting vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator. For instance, In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...

where the denominator 42 was used because lcm(21, 6) = 42.

Calculating the least common multiple

If a and b are not both zero, the least common multiple can be computed by using the greatest common divisor (gcd) of a and b: In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

Thus, the Euclidean algorithm for the gcd also gives us a fast algorithm for the lcm. To return to the example above, In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... Flowcharts are often used to graphically represent algorithms. ...

Because (ab)/c = a(b/c) = (a/c)b, one can calculate the lcm using the above formula more efficiently, by first exploiting the fact that b/c or a/c will be easier to calculate than the quotient of the product ab and c, because the fact that c is a factor of both a and b entails that in either fraction, a/c or b/c, one can completely cancel the c. This can be true whether the calculations are performed by a human, or a computer, which may have storage requirements on the variables a, b, c, where the limits may be 4-byte storage - calculating ab may cause an overflow, if storage space is not allocated properly.

Using this, we can then calculate the lcm by either using:

or

Done this way, the previous example becomes:

Even if the numbers are large and not quickly factorable, the gcd can be calculated quickly with Euclid's algorithm. In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...

Alternative method

The unique factorization theorem says that every positive integer number greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... A composite number is a positive integer which has a positive divisor other than one or itself. ...

For example:

Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5.

This knowledge can be used to find the lcm of a set of numbers.

Example: Find the value of lcm(8,9,21).

First, factor out each number and express it as a product of prime number powers. In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

The lcm will be the product of multiplying the highest power in each prime factor category together. Out of the 4 prime factor categories 2, 3, 5, and 7, the highest powers from each are 2³, 3², 5⁰, and 7¹. Thus,

Viewing this method via Venn diagrams

One can find the least common multiple of two numbers by using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram. A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ... In mathematics, the integer prime-factorization (also known as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. ...

Here is an example:

48 = 2 × 2 × 2 × 2 × 3,

180 = 2 × 2 × 3 × 3 × 5,

and what they share in common is two "2"s and a "3":

This also works for the greatest common divisor (GCD), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the GCD of 48 and 180 is 2 × 2 × 3 = 12. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

The 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 ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... UFD redirects here, but this abbreviation can also mean USB flash drive, an electronic device. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...

See also

• Lowest common denominator
• Anomalous cancellation

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ... An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. ...

External links

• Online LCM calculator
• Online lcm calculator
• Online LCM calculator
• LCM Quiz
• LCM and GCF solvers, work shown These solvers use factorization algorithm described in wikipedia.