MHB Greatest Common Divisor: Applications in Tiling and Number Theory

[matqkks]

Are there any real life applications of greates common divisor of two or more integers?

 

[Nono713]

Absolutely. It is used basically everywhere in number theory, and, therefore, in all applications of number theory. If you mean applications depending only and directly on the greatest common divisor, then, for instance:

If you have a rectangle of dimensions $\text{W} \times \text{H}$ metres - integer dimensions, obviously - then the largest square that can tile the rectangle perfectly has side length $\gcd \left ( \text{W}, \text{H} \right )$.

This has - on some level - applications in everyday tiling: if you want your tiling to correctly fit the floor, you want to make sure you meet the condition above, else it will look odd as the edge tilings won't fit and will need to be cut down. This depends on the dimensions of the floor that is to be tiled. The same reasoning can be applied to more complex tilings (which are often compositions of simple tilings of various shapes) to make sure they all fit together and repeat nicely.

That said, if you are looking for more concrete, "serious" applications, I'm not sure you'll find any to your liking. The greatest common divisor isn't a particularly broad concept but rather a mathematical building block, and your question would then be akin to "what are real life direct applications of addition" which I am sure you can agree is difficult to answer (the best answer being, ultimately, "everything, on some level").​