Least common multiple
The smallest common multiple of two numbers is the smallest positive integer that is a multiple of both numbers. We can also calculate the least common multiple for more numbers, then the least common multiple is the smallest positive integer that is a multiple of all the numbers.
For example, consider two numbers 3 and 4. We are looking for a number that is a multiple of both 3 and 4.
- The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, ...
- The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, ...
We can see that in a series, some of the multiples are the same. For example, the number 24 is a multiple of both the number 3 and the number 4. But we are not looking for any common multiple, we are looking for the multiple that is the smallest. So the smallest multiple is the number 12.
How to calculate the smallest common multiple
The easiest way is to find the greatest common divisor of the two numbers and then calculate the least common multiple using the formula:
$$\mbox{ NSN }(x, y) = \frac{x \cdot y}{\mbox{ NSD }(x, y)}$$
where NSN(x, y) is the greatest common multiple of the numbers x and y and NSD(x, y) is the least common divisor of the numbers x and y. So for an example, let's take the numbers 50 and 35. Their greatest common divisor is the number 5. So we can calculate that
$$\mbox{ NSN }(50, 35) = \frac{50 \cdot 35}{5} = \frac{1750}{5}=350$$