If has 58 trailing zeros, what is n?

where

I don't know what to do next.

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- Jun 19th 2010, 10:13 AMdwsmithn!
If has 58 trailing zeros, what is n?

where

I don't know what to do next. - Jun 19th 2010, 10:29 AMundefined
Actually you can write since the terms eventually all become 0.

We can get an immediate upper bound: if n=59*5, then the first term will be greater than 58. Since 59*5 - 1 < 625, clearly we have just the first three terms:

This is easy to play around with; 58/5 is about 12, so as an estimate we can subtract about 10*5 from our upper bound to get 59*5-51 = 244. Plugging in results in 48 + 9 + 1 = 58, as desired. So . - Jun 19th 2010, 08:07 PMsimplependulum
Or let

as a representation in base 5 , since it does not exceed

Therefore ,

and ranges from to .

.