To start solving this problem, we first need to determine how many passengers are in each set of three consecutive cars. Given that every three consecutive cars have exactly 59 passengers, we can divide the total number of passengers by the number of sets of three cars. With 14 cars, there are 12 sets of three consecutive cars (1,2,3; 2,3,4; 3,4,5; and so on until 12,13,14). So, we divide 276 (the total number of passengers) by 12 (the number of sets of three consecutive cars), giving us an average of 23 passengers per set of three cars.

However, we know this cannot be the case because the problem states that every three consecutive cars have exactly 59 passengers, not 23. Therefore, the condition that every three consecutive cars have exactly 59 passengers is not possible with the total of 276 passengers divided into 12 sets of three consecutive cars.

Therefore, it seems there might be a mistake in the problem statement. As such, I am unable to determine which car(s) Jack may ride in.

AI made many mistakes. First, the total number of passengers is 277, not 276. Second, adding consecutive cars 1-2-3 with 2-3-4 is absurd because this way we count passengers from the cars 2 and 3 twice.

Denote the number of passengers in the second car as a, then the third car has a passengers while the first car has 59 - 2a passengers.

Consecutively, the fourth car must have a passengers again as well as the
fifth, and the pattern is repeated:

59-2a, a, a, 59-2a, a, a, 59-2a, a, a, 59-2a, a, a, 59-2a, a.

The total is (59-2a)*5 + a*9 = 295 - a = 277, so a = 18 and 59-2a = 23.
Thus, the cars has the following quantities of passengers and the car
numbers:

23 (1), 18 (2), 18(3), 23(4), 18 (5), 18 (6), 23 (7), 18 (8), 18 (9), 23 (10),
18 (11), 18 (12), 23 (13), 18 (14),

so the sums are

24, 20, 21, 27, 23, 24, 30, 26, 27, 33, 29, 30, 36, 32.

The only prime numbers here are 23 and 29, and their order numbers are
**5** and **11**.

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