Question

Vector Prime Number Stuff (ok I'm here this time to say what I was going to say an hour or two ago)

Answer 1

For every pair of 2D vectors there is a GCD and LCM that gives us back a new vector with either the smallest or largest components of the two vectors respectively.

Now what if we have a 3D vector or higher, what can we do there? We essentially have a new value we can consider like this, which I'll call the mcf and explain in a second.

\[\large \gcd(a,b,c) lcm(a,b,c) mcf(a,b,c)=a*b*c\]

We can easily solve for it every time because the other values are known to us.
\[\large mcf(a,b,c)=\frac{a*b*c}{\gcd(a,b,c) lcm(a,b,c) }\]

Although it looks complicated, finding this new function mcf(a,b,c) really just gives us kind of like a median of common factors. So if we have these numbers:

\[a=2^33^2 \\ b=2^43^1 \\ c=2^23^4\] then we have \[\gcd(a,b,c)=2^23^1 \\ lcm(a,b,c)=2^43^4 \\ mcf (a,b,c)=2^33^2\]

And really all we do to find the mcf is look at the middle terms.

What is the geometric interpretation of this?

Answer 2

However in this instance a is also the mcf, if it isn't then it will always form a circuit instead of "poking" a hole in the center of the rectangle, so it might look like this: