Given x!, Find x

Date: 12/21/2002 at 10:16:04
From: John Macloud
Subject: Factorials

If the value of x! is given, how do we find the value of x?

Example: x!=5040  find x

Date: 12/21/2002 at 15:11:46
From: Doctor Achilles
Subject: Re: Factorials

Hi John,

Thanks for writing to Dr. Math.

The best way I can think of is a kind of guess-and-check.

Here's how I'd solve it:

If n!=1, then n=1 or n=0.

If n!>1, then divide by 2. After you do that, if you are left with 1, 
then n=2; otherwise repeat for 3.

So for your problem, x!=5040 which does not equal 1, we have to divide 
by 2. That gives us 2520.

2520 does not equal 1, so we have to divide by 3. That gives us 840.

840 does not equal 1, so we have to divide by 4. That gives us 210.

210 does not equal 1, so we have to divide by 5. That gives us 42.

42 does not equal 1, so we have to divide by 6. That gives us 7.

7 does not equal 1, so we have to divide by 7. That gives us 1.

1 equals 1, so x must have been 7.

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/25/2002 at 02:05:01
From: John Macloud
Subject: Thank you (Factorials)

Thanks very much for your quick answer, it really helped me out. 

Thanks,                 
John Macloud