If you're still curious, there are a few ways:
The most old-fashioned is an injection locked oscillator. The oscillator is typically a blocking oscillator or relaxation oscillator type, which has a voltage or current that charges slowly, until approaching, and then crossing a threshold. When that happens, it fires a pulse and resets the cycle. If a small perturbation is added to the 'charge' signal, or subtracted from the threshold -- same thing, then when that perturbation is high (near a peak), the oscillator is more likely to reset early, and in phase with that signal. Depending on the unperturbed frequency, and the magnitude of perturbation, many oddball ratios can be had, normally something like a ratio of p/q with p+q < 10 or so. They can be chained to get very large ratios, of course.
A few examples are: to generate TV scan lines (NTSC TV is 525 lines per frame, interlaced alternate fields; meaning, at the field rate of 60Hz, each field is alternately 262 or 263 lines -- a toggle flip-flop would be needed to select which sequence occurs on alternate cycles, or an extra line would be inserted or removed somehow). The horizontal sync was, in turn, derived from the colorburst frequency (when color came along), with a ratio of... 228? Another application was in somewhat cheaper organs in the 60s, which typically used tube LC oscillators for the highest octaves, and germanium transistor flip-flops (essentially injection locked astable multivibrators) to generate the lower octaves. (Between the germanium transistors, Z5U ceramic capacitors, and ambient temperatures produced by the tube power amp, among other things, even this 1:2 ratio apparently wasn't terribly stable!)
Apparently, the technique is still in use today for extremely high frequency prescalers (>30GHz) in frequency counters; at least, I've seen a few research articles on it. No idea what uses this commercially.
Straight multiplication is easy. Often used in radio up-converters. You take a tuned RF amplifier, set it for a particularly nasty bias condition so it generates a lot of distortion, and tune the output for one of those harmonics. Ratios up to 5:1 are practical, and much larger non-prime numbers are achieved through cascading. An example might be a reactance modulated FM/PM source: because the phase shift is nominal (within +/-90 degrees), the deviation is very small (fractional percent), so for the FM to be of practical magnitude (+/-10kHz, say), it has to be multiplied significantly. You might have a crystal oscillator at 1MHz, reactance mod, then multipliers to take that up to, say, the ~500MHz amateur band (576MHz..?), where the deviation is acceptable as "narrow band FM". (Commercial wide band FM (100kHz deviation) is produced through different means; originally with difficulty, in rather elaborate systems that mechanically servoed the drifty frequency; later, using a fantastic tube called a phasitron; then with varactor control and PLLs; digital synthesis is probably used today.)
Other than that, the canonical digital method is a VCO phase-locked to a reference. The reference is divided by q with a digital counter, and the VCO output is divided by p with another. The counter outputs are what the PLL is comparing. Thus, the VCO runs at p/q times the input.
Tim
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