Derive cyclotron and synchrotron frequency

[Aziza]

I am trying to find out how to derive the cyclotron and synchrotron frequency, but everywhere I look it is derived by saying that the force of the magnetic field is not qvB, but (v/c)qB. What on Earth is this c? I am guessing it means speed of light but then it is not the case that (v/c)qB = (mv^2)/r...but that is what is says everywhere for this derivation. I have attached a snapshot of a website deriving it this way as an example.

 

[nasu]

It's a matter of units.
The formula with c in it is the expression of Lorentz force in CGS units.
The one that you are more familiar with is in SI units.