How to find large prime factors without using computer?

What is the largest prime factor of the number 600851475143 ?
This is the third problem of Project Euler.

How to approach this mathematically (without computer programming) ?

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Without using a computer? Are you for serious? – Cocopuffs Jun 22 '12 at 9:56
Here's a challenge: Try to think of a title that would be more broadly applicable to all questions on this site than the one you chose. – joriki Jun 22 '12 at 10:03
The topic of Project Euler questions has been discussed on meta before. Voting to close (although I don't know what reason to catalog this under). – Asaf Karagila Jun 22 '12 at 10:28
@Asaf I've added the project-euler tag, since it is explicitly mentioned in the question. If you don't think this is correct, feel free to remove the tag. (Based on tag wiki, I am not sure whether the intended use of this tag is for everything related to PE, or just for question which are directly copied from there. It seems that this question is different from the question at PE, if PE expect program ass a solution.) – Martin Sleziak Jun 22 '12 at 10:38

This would work for small prime factors...however, I don't know of any divisibility rules for say the prime $73$. You would have to mess about to derive one. – fretty Jun 22 '12 at 10:48
Most divisibility rules for small numbers depend on nice things about the multiplicative orders of $10$ mod $n$. If $n$ gets bigger then chances are the order gets bigger and so nothing much is gained by using such a divisibility rule. – fretty Jun 22 '12 at 10:54