The technique that produces the seventy-two names is arithmetical, and it can be checked by anyone with a Hebrew Bible.
Exodus 14:19, 14:20 and 14:21 are three consecutive verses in the account of the crossing of the sea. Each contains, in the received Hebrew consonantal text, exactly seventy-two letters. That equality is the whole basis of the system, and it is a real property of the text rather than a claim about it.
The three verses are then read in a boustrophedon, literally an ox-turning, the pattern of a plough moving down one furrow and back up the next. The first verse is taken left to right in the sequence of its letters, the second in reverse, the third forward again. Taking the first letter of each verse in that arrangement gives a group of three letters; the second letter of each gives the next; and so on to the seventy-second. The result is seventy-two three-letter groups.
Those groups are the Shem ha-Mephorash in the sense the phrase carries in this tradition: the divine name, divided. Each is then made pronounceable by a suffix, either -el or -yah, both of which are divine elements in ordinary Hebrew names, and the seventy-two become names of angels.
Two features of the arrangement are worth noting. It is symmetrical and it is exhaustive: every letter of all three verses is used, once, with nothing left over and nothing repeated. And it is not stated in Exodus. The text of the crossing narrative gives no hint that it is to be read this way. The technique is imposed on it, by later readers, for reasons the passage itself does not supply.