The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, and so on. How many of the first 12 such sums are also prime?
I need to find the first 12 prime sums and check if they are prime. The first prime sum is $2$, which is prime. The second prime sum is $2 + 3 = 5$, which is also prime. The third prime sum is $2 + 3 + 5 = 10$, which is not prime, since it is divisible by 2 and 5. The fourth prime sum is $2 + 3 + 5 + 7 = 17$, which is prime. The fifth prime sum is $2 + 3 + 5 + 7 + 11 = 28$, which is not prime, since it is divisible by 2 and 7. The sixth prime sum is $2 + 3 + 5 + 7 + 11 + 13 = 41$, which is prime. The seventh prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 = 58$, which is not prime, since it is divisible by 2 and 29. The eighth prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77$, which is not prime, since it is divisible by 7 and 11. The ninth prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100$, which is not prime, since it is divisible by 2, 4, 5, 10, and 20. The tenth prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129$, which is not prime, since it is divisible by 3 and 43. The eleventh prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 160$, which is not prime, since it is divisible by 2, 4, 5, 8, 10, and 20. The twelfth prime sum is $2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 = 197$, which is prime. Therefore, out of the first 12 prime sums, only 5 are also prime: $2$, $5$, $17$, $41$, and $197$. 