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This question already has an answer here:

Is there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do factorials and factorial roots (for lack of a better term)?

Here is an example: 5! = 120. Is there a way I can work out the number that must be factorialed (??) to give the answer 120?

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marked as duplicate by Daniel Fischer Jan 22 '17 at 13:35

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You can just divide the "answer" by consecutive positive integers, and when the result is 1, the last number you divided by is the number that the "answer" is factorial of. For example: 120 / 2 = 60, 60 / 3 = 20, 20 / 4 = 5, 5 / 5 = 1, so the number that 120 is the factorial of is 5.

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    $\begingroup$ This would probably be the simplest and quickest way to do it (maybe not by hand for large numbers). If at some stage, you divide by $k$ exactly, and at the next step you cannot divide by $k+1$ exactly, then, clearly, you do not have a factorial number. $\endgroup$ – digital-Ink Jul 17 '12 at 11:43
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    $\begingroup$ @digital-Ink Then I'd like to call these numbers perfect factorials. $\endgroup$ – Simply Beautiful Art Dec 27 '15 at 15:38

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