That's how harmonics work. Harmonics are basically amplification of oscillation where two or more oscillation periods coincide. They can't occur at fractional ratios, because the oscillations have to be in-phase, IE, whole-number multiples of each other.
Think about making waves in a bathtub: when you splash at a regular frequency, the waves get bigger because each new impulse coincides with the crest or trough of the existing waves.
I just realized harmonics are really difficult to explain.
Our hypothetical 12-bladed fan produces a harmonic oscillation for every balanced pair of blades. Essentially, axes of symmetry on a rotating mass create oscillations at regular intervals. Non-prime numbers of blades will result in more axes of symmetry.
So the 12-blade fan oscillates at w(rate of rotation)/1, w/2, w/3, w/4, w/6, and w/12.
A 13-bladed fan will only oscillate at w/1 and w/13, and so will seem much quieter. The two oscillation frequencies do not have any common factors, so no harmonic amplification, so less vibration, so less noise.
Clear as mud?