Get a smallest three digit prime number which has All prime digits Forms a prime number with its sum of first two digits. Forms a prime number with its sum of last two digits.

I think the smallest one is 523. EDIT :: The reason is : We have only 4 prime digits : 2, 3, 5 and 7. We can have 4C2 = 6 different combinations, taking 2 at a time. Out of these 6, only 2 have prime sums. They are 2,3 and 2,5. So, the number will be 325 or 523 as 2 should be common to 3 and 5. So, obviously 523 is the answer.

Correct and also I have a bit different logic. 2 Needs to be at middle because adding two odd number would always be even and so unprime So now we have _2_ Now we have 3 which can be at last place because 2 and 5 would make the number unprime 7 would make the sum unprime _23 Now and 3, 5 at first place are the only option and so 323 is not a prime number. I added smallest just to add confusion but the solution is unique.