When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$?
So, I know that one of $p_1, p_2, p_3$ must be even, and the only even prime is $2.$ We can WLOG let $p_1=2,$ so $p_2+p_3=102$ (because $p_1+p_2+p_3=104$).
Where I'm stuck:
How do you proceed to find all the positive integers for which $p_1 + p_2 + p_3 = a+b+c$?
Any hints/solutions would be appreciated! Thanks.