If the frequencies are integers, like 213 or 5193 cycles per second, then that means that each frequency has an even number of cycles in a one second interval. In other words, each waveform has a period such that one second is a multiple of that period. Therefore, in a mixture of such tones, one second is a multiple of each of their periods, which makes it a common multiple (and not necessarily the lowest common multiple). If we take a period that is a common multiple of the waveforms, and repeat that period, we get a function that is indistinguishable from a continuation of the original. Or, in other words, any phase shift of a signal by a multiple of one second is indistinguishable from the unshifted signal, if the signal contains only integral frequencies.
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