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) ?
Well the point of Project Euler is to program. This is a problem that you could solve by hand but it would take you quite a while.
Factorisation is not a simple thing to do by hand for numbers this big (unless you have some special insight into divisibility by certain special primes).
Just use the bog standard test, square root and check divisibility of primes upto this. Once you find one, divide out and start again. In the end you will have a number that you cannot do this process to. This will be the largest prime factor.