LCM of Decimals. Making sense of non-sensible results.

[BigBeachBanana]

I've recently come across an article from geeksforgeeks on how to find the LCM of decimal numbers. The suggested solution is as follows:

Find the LCM of 3, 2.7, 0.09.
Step 1: Convert to integers by multiplying by 100 -> 300, 270, 90
Step 2: Find the LCM via the prime factorization method -> 2700
Step 3: Divide by number of decimals moved in step 1 -> 2700/100 = 27.

This result is sensical as the example is conveniently chosen as a multiple of 3's. However, applying the same steps to something such as 27.2, 28.612, and 29.36 the result is 71,404,107.20

This result appears silly but seemingly correct. How to make sense of this?

 

[blamocur]

However, applying the same steps to something such as 27.2, 28.612, and 29.36 the result is 71,404,107.20

This result appears silly but seemingly correct. How to make sense of this?

Why silly? I am guessing LCM (27200, 28612, 29360) = 71,404,107,200, correct?
I think they are are using the fact that [imath]LCM (a\cdot x, a\cdot y, a\cdot z) = a \cdot LCM (x,y,z)[/imath]

 

[Dr.Peterson]

I've recently come across an article from geeksforgeeks on how to find the LCM of decimal numbers. The suggested solution is as follows:

Find the LCM of 3, 2.7, 0.09.
Step 1: Convert to integers by multiplying by 100 -> 300, 270, 90
Step 2: Find the LCM via the prime factorization method -> 2700
Step 3: Divide by number of decimals moved in step 1 -> 2700/100 = 27.

This result is sensical as the example is conveniently chosen as a multiple of 3's. However, applying the same steps to something such as 27.2, 28.612, and 29.36 the result is 71,404,107.20

This result appears silly but seemingly correct. How to make sense of this?

The first question you need to ask is, what is the definition of LCM for non-integers? Only when that's been done can you really say if it makes sense, or doesn't. The definition has to be this:

A multiple of a (not necessarily integer) number is the result of multiplying that number by an integer.​

The LCM of a set of numbers is the smallest number that is an (integer) multiple of each of them.​

Now, with a definition, we can determine whether the method makes sense. If the numbers are all rational, then we can write them with a common denominator, and it makes sense that the LCM will be the LCM of the numerators, over that same denominator. And that's what they are doing with their decimals (using a power of ten as common denominator).

 

[BigBeachBanana]

The first question you need to ask is, what is the definition of LCM for non-integers? Only when that's been done can you really say if it makes sense, or doesn't. The definition has to be this:

A multiple of a (not necessarily integer) number is the result of multiplying that number by an integer.​

The LCM of a set of numbers is the smallest number that is an (integer) multiple of each of them.​

Now, with a definition, we can determine whether the method makes sense. If the numbers are all rational, then we can write them with a common denominator, and it makes sense that the LCM will be the LCM of the numerators, over that same denominator. And that's what they are doing with their decimals (using a power of ten as common denominator).

I started the process out wrong. This is what I was missing.