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The least common multiple is useful when adding or subtracting 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:

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

1 Efficient calculation

The formula

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

Because that (ab)/c = a(b/c) = (a/c)b, one can calculate the lcm using the above formula more efficiently, by firstly exploiting the fact that b/c or a/c may be easier to calculate than the quotient of the product ab and 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:

2 Alternative method

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

3 External links

• Online lcm calculator
• LCM Quiz