Math Is Fun Forum

How can I get going with this question? Thxes

Is it correct that:

I would think that, but my calculator comes up with

. Is it right?

Is there any way to evaluate log to the base 10 of any number without a calculator? I don't really care if it's by trial and error, similar to the trial and error used in finding roots, but is there an algorithm or method to doing it? Thanks.

1. Express the number 1/2006 as the sum of the reciprocals of distinct integers of size less than 2006 - that is,

,

where -2006 <

< 2006 for all i = 1,2, ... , n.

Well... all I really know is that the denominators must multiply to a multiple of 2006... so must be a factor of a multiple of 2006... but what multiple...?
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2. Find all positive integers less than 10,000 which are equal to five times the product of their digits.

Guess that means:

I've already worked out that d = 5. This is because 1) the number must be a multiple of 5, and 2) no digit may be zero.
Next, every digit must be odd. This is because if any even number is multiplied by 5, the last digit of the product will be 0, and this is not allowed.
So we have:

, where

Don't know how to go further...
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These are the final two 'puzzle' type questions I'm having trouble with, in addition to the ones I've already posted... preparation for a math test on Wednesday... thanks.

Find all real values of x which fit the given equation:

I've tried graphing it and there are 5 zeros, but how do I find them algebraically?