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. If there is no such positive integer, e.g., if a = 0 or b = 0, then lcm(a, b) is defined to be zero.

The least common multiple is useful when adding or subtracting vulgar fractions, because it yields the lowest common denominator. Consider for instance

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

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}={21\cdot 2}=42.</math>

1 Efficient calculation
- 1.1 A way to remember to cancel before multiplying
2 Alternative method
3 See also
4 External links

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Efficient calculation

The formula

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

is adequate to calculate the lcm for small numbers using the formula as written.

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 multiple 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)=a\cdot\left({b\over\operatorname{gcd}(a,b)}\right).\,</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.

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A way to remember to cancel before multiplying

Those who have taught elementary mathematics sometimes find it frustratingly difficult to get students to remember to cancel before multiplying. The following way of arranging this algorithm has the virtue of making that step impossible to forget in this case (essentially by making it unnecessary to remember). We illustrate it via the example of finding lcm(12, 8).

First reduce the fraction to lowest terms: <math>{12 \over 8} = {3 \over 2}.</math>

Then "cross-multiply": <math>12\times 2 = 8\times 3.\,</math>

The product 12 × 2 = 8 × 3 = 24 is the lcm.

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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.

We can use this knowledge to easily find the lcm of a group of numbers.

For example: Find the value of lcm(45, 120, 75)

The lcm is the number which has the greatest multiple of each different type of atom. Thus

<math>\operatorname{lcm}(45,120,75) = 2^3 \cdot 3^2 \cdot 5^2 = 8 \cdot 9 \cdot 25 = 1800. \,\!</math>

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