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Division by e (Posted on 2020-03-25) Difficulty: 4 of 5
Let [x] denote the closest integer to x.

Find the last 5 digits of [100000!/e]

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (2 votes)

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Guess not proof | Comment 1 of 7
BY "the last five digits", I assume that what is meant is the digits in the ones thru ten-thousands place in decimal notation. 

I note that, using Excel, ever increasing powers of 10 divided by e begin to have more and more zeros just to the left of the decimal point.  Since 100000! has many many 10s in the product, it will have many many consecutive zeros to the left of the decimal point also.  So the answer might be:

00000

However, we are all aware of the limitations of Excel, and it might be that the representation of "e", which is EXP(1), is truncated, which would mean that the real value of "e" is a bit larger.  If that's the case (and I have no proof), then the answer to this still would be (closest integer) problem would still be 0000.

Help anyone?  I suspect there is a much nice proof that could be offered.

  Posted by Kenny M on 2020-03-26 09:53:50
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