There are several processes that allow us to calculate the L.C.M (Lowest Common Multiple) between two or more numbers. We could start by creating two lists containing all the multiples of both numbers and we would stop when we found the common number between them. But this is not a practical process! The best method is to start by decomposing each one of the numbers according to a product of prime numbers. This process is called to factorize and it is very practical, because it can be used not only to calculate the G.C.D but also the L.C.M. Let´s suppose we want to calculate the L.C.M between 168 and 180.

After having decomposed the numbers in prime factors, we can notice that `168 = 2^3 xx 3 xx 7` and `180 = 2^2 xx 3^2 xx5`. The following step is to calculate the product between all the common factors having the biggest exponent as well as the one of non-common factors. In this case we would have `2^3 xx 3^2 xx 5 xx 7 = 2520`. Thus, we arrive to the conclusion that the Lowest Common Multiple between 168 and 180 is 2520.

There are several examples of real problems where the calculation of the L.C.M becomes useful. Let’s imagine that a circular running track in which two runners are competing and they run round the track several times. Now, let’s suppose that it takes one of them 168 seconds to complete a full lap of the track while it takes 180 to the other. We know they leave at the same time: when will they join together at the start? In this example, as the L.M.C of these two numbers is 2520, it allows us to say that they will join together at the start in 2520 seconds (42 minutes). But, do not forget that the faster one has already completed more laps.

We call your attention to the fact that the L.C.M is not only calculated between two numbers. We can calculate the L.C.M of 2, 3, 4 or more numbers! Having this in mind, if we have two natural numbers and we start writing multiples of each one of them, we can assure you that, at a certain point, we will reach a common multiple (the worst that can happens is that we can always multiply one number by the other). The first common multiple we will find is called the L.C.M.

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