Tips for Prime Factorization of a Given Large Interger This may be a slightly silly question, but are there any tips for prime factorization when slight hints are given?
For example, if you were not possesed of pen, paper, or calculator, and somebody asked you to prime factor $294020099$ with the three hints given below, is it possible there are some ways that you could do it within a short amount of time?
Hint $1$.The prime factors are between $500$ and $900$
Hint $2$.The number $294020099$ has three prime factors 
Hint $3$.The sum of all prime factors are equal to $2049$. 
While I am aware that there are some computer algorithims, actually doing them in your head seemed difficult. Any tips would be appreciated. 
 A: Without a pen and a paper it is really hard; but one could solve it by hand, without a computer as follows. First we try
to find the smallest divisor $d$ of $294020099$ with $d\ge 500$. One would quickly see that $d=503$, because the division by $501$ leaves a remainder, and $500$, $502$ are even, hence impossible.
 Then one had to show that $503$ is indeed prime, and consider the cofactor $584533$. Now it gets harder, but not impossible. 
A: Let $a,b,c$ denote the three prime factors.
Each prime factor must end with either $1$, $3$, $7$ or $9$.
In addition, each one of the following two conditions must hold:


*

*$a\times b\times c\equiv9\pmod{10}$

*$a   +   b   +   c\equiv9\pmod{10}$


A quick check reveals that WLOG:


*

*$a\equiv3\pmod{10}$

*$b\equiv7\pmod{10}$

*$c\equiv9\pmod{10}$


So the general algorithm would be:


*

*Choose $a$ from the following lists:


*

*$503,533,563,\dots,833,863,893$

*$523,553,583,\dots,823,853,883$


*Choose $b$ from the following lists:


*

*$517,547,577,\dots,817,847,877$

*$527,557,587,\dots,827,857,887$


*Choose $c$ from the following lists:


*

*$509,539,569,\dots,839,869,899$

*$529,559,589,\dots,829,859,889$


*If $a+b+c=2049$, then check if $a\times b\times c=294020099$

