Lowest Common Multiple (LCM)

The three concepts: lowest, common and multiple will be easier understood if explained in reverse order, i.e. multiple, common, lowest.

Multiple

Remember the times table?
2 × 1 = 2          3 × 1 = 3          4 × 1 = 4
2 × 2 = 4         3 × 2 = 6         4 × 2 = 8
2 × 3 = 6          3 × 3 = 9         4 × 3 = 12
2 × 4 = 8         3 × 4 = 12       4 × 4 = 16
2 × 5 = 10        3 × 5 = 15        4 × 5 = 20
Now, the solutions under a particular number are its multiples. So,
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

Common

When considering the multiples of two numbers, the ones that appear for both are the “common multiples.” For instance,
Common multiples of 2 and 3 = 6, 12, 18, 24, 30, …
Common multiples of 2 and 4 = 4, 8, 12, 16, 20, …
This also applies for more than two numbers. Hence,
Common multiples of 2, 3 and 4 = 12, 24, 36, 48, 60, …

Lowest

Of all the common multiples, which one is the smallest, minimum, least? Therefore,
Lowest common multiple of 2 and 3 = 6

LCM of 2 and 4 = 4
LCM of 2, 3 and 4 = 12

But, does that mean that to determine the LCM of 2, 3 or 4 numbers, all their multiples have to be listed, then the common ones are identified before picking the lowest? Certainly not. There are standard ways of finding the LCM. However, the method or technique to use depends on the RELATIONSHIP between the numbers under consideration. Generally, numbers are related as follows:

  1. Same e.g. 2 and 2, 3 and 3, 4 and 4.
  2. One being a multiple of the other e.g. 2 and 4, 3 and 6, 4 and 8.
  3. Neither (i) nor (ii) above e.g. 2 and 3, 4 and 5, 7 and 10.

When numbers are the same, their LCM is simply the number itself. For example, what is the LCM of 5 and 5?

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
Multiples of second 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
Common multiples of both = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
As a result, the LCM of 5 and 5 = 5.
In the field of inheritance, this is called TAMATHUL i.e. the same.

If two numbers are related such that one is a multiple of the other, their LCM is the higher number. For instance, what is the LCM of 3 and 6?
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Common multiples of 3 and 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, …
Thus, the LCM of 3 and 6 = 6.

Notice that 6 is a multiple of 3, which is why the common multiples of 3 and 6 are actually the multiples of 6! Hence, their LCM is simply the LCM of 6 since it is the higher number. Similarly, the LCM of 3 and 21 is 21 and the LCM of 6 and 42 is 42. This phenomenon is referred to as TADAKHUL, meaning multiple.

In a situation whereby the numbers under consideration are “neither,” i.e. are not the same and one is not a multiple of the other, the easiest way to determine their LCM is to MULTIPLY them. Example, what is the LCM of 2 and 3?
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
Common multiples of 2 and 3 = 6, 12, 18, 24, 30, 36, 42, …
So, LCM of 2 and 3 = 6.

Likewise, LCM of 4 and 5 is 20 and LCM of 7 and 12 is 84. This is called TABAYIN in inheritance literature. I translate it as “parallel.” The fourth relationship between numbers shall be unveiled in due course. Meanwhile, knowledge of these relationships is the SECRET of determining the base number and by implication the whole of inheritance arithmetic! That is why a lot of space is being devoted to explain these basics.

Let’s complicate the problem. How do we find the LCM of 3 or more numbers? First, pick any 2 numbers and find their LCM. Call this LCM ‘X’. Next, pick a 3rd number. Determine the LCM of this 3rd number and ‘X’. Call this new LCM ‘Y’. Then find the LCM between ‘Y’ and the 4th number. And the process continues. The final LCM is the LCM of all the numbers.

Example, what is the LCM of 2, 5 and 10? Considering the first two numbers 2 and 5, they are parallel, so multiply them to get their LCM. It’s 10. But this solution, 10 and the 3rd number, 10 are the same. And the LCM of similar numbers is that number. Thus, the LCM of 2, 5 and 10 is 10.

Alternatively, if 2 and 10 were considered first, 10 is a multiple of 2 so the higher number, 10 is chosen as the LCM. Incidentally, the solution, 10 is also a multiple of the 3rd number, 5. As a result the higher number, 10 is picked and that is the LCM of the 3 numbers. The LCM of the 3 numbers will equally be 10 if 5 and 10 are taken first. That is the beauty of Mathematics. It does not lie!

These methods of finding LCM are also applicable to fractions. But in their case, only the DENOMINATORS are considered. For instance,
LCM of 2/3 and 1/3 = 3           Both denominators are the same.
LCM of ½ and 1/6 = 6           6 is a multiple of 2, so pick the higher one.
LCM of ¼ and 1/7 = 28         4 and 7 are parallel, so multiply them.

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