Each of A, B, C, A+B*C, B+C*A and C+A*B is a prime number.
Find the possible remainders when A+B+C is divided by 6.
All prime numbers over 3 are 1 or -1 (mod6), therefore none of the binoms of A+BC structure will be prime if A*B*C is an odd number.
It is easy to see that one (and not more than one!) of(A,B,C) MUST BE 2, so ABS(A*B*C) will be 2, i.e.
The possible remainders when A+B+C is divided by 6 are either 2 or 4
Example: 2,5,7 generate 37,19,17 ;
2*5*7=70 whereas 70 divided by 6 gives 11 and remainder 4.