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Least common multiple

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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, a and b divide it without 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. 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,

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

1 Calculating the least common multiple
2 Alternative method
- 2.1 Another way of viewing this method
3 The lcm in commutative rings
4 See also
5 External links

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:

<math>\operatorname{lcm}(a,b)=\frac{a\cdot b}{\operatorname{gcd}(a,b)}.</math>

Thus, the Euclidean algorithm for the gcd also gives us a fast algorithm for the lcm. To return to the example above,

<math>\operatorname{lcm}(21,6)

={21\cdot6\over\operatorname{gcd}(21,6)} ={21\cdot 6\over 3}={126\over 3}=42.</math> 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:

<math>\operatorname{lcm}(a,b)=\left({a\over\operatorname{gcd}(a,b)}\right)\cdot b</math>

<math>\operatorname{lcm}(a,b)=\left({b\over\operatorname{gcd}(a,b)}\right)\cdot a</math>

Done this way, the previous example becomes:

<math>\operatorname{lcm}(21,6)={21\over\operatorname{gcd}(21,6)}\cdot6={21\over3}\cdot6=7\cdot6=42.</math>

Even if the numbers are large and not quickly factorable, the gcd can be calculated quickly with Euclid's algorithm.

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. 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.

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,

<math>\operatorname{lcm}(8,9,21) = 2^3 \cdot 3^2 \cdot 5^0 \cdot 7^1 = 8 \cdot 9 \cdot 1 \cdot 7 = 504. \,\!</math>

Another way of viewing this method

One can also find the LCM (least common multiple) by finding the prime factorization of each number (by using the factor tree). Once that is found, you can create and fill in a Venn diagram with a circle for each number. To find the LCM, just multiply all of the prime numbers in the diagram. Here is an example: Note: This also works for the greatest common factor (GCF). Except, instead of multiplying all of the numbers in the Venn diagram, you only multiply the prime factors that are in common with all of the numbers. So the GCF of 12 and 36 would be 2 × 2 × 3 or 12.

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 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.