Four prime numbers are written in ascending order. The product of the first three primes is 385, and the product of the last three primes is 1001. What is the value of the fourth (largest) prime?

Introduction:
This question combines prime factorization with logical ordering. You are told that four prime numbers are in ascending order, and you are given two overlapping products: the product of the first three and the product of the last three. The challenge is to factor each product into primes and then match the overlapping primes to identify the full set of four and the largest one.

Given Data / Assumptions:

• The four primes are p1 < p2 < p3 < p4.
• p1 * p2 * p3 = 385.
• p2 * p3 * p4 = 1001.
• All numbers mentioned are positive primes.

Concept / Approach:
Factor 385 and 1001 into their prime factors. Since p2 and p3 are common to both products, they will appear in both factorizations. The extra prime in the first factorization will be p1, and the extra prime in the second factorization will be p4. This approach quickly reveals all four primes in ascending order.

Step-by-Step Solution:
Prime factorize 385: 385 = 5 * 7 * 11.So p1, p2, p3 must be 5, 7, 11 in ascending order.Prime factorize 1001: 1001 = 7 * 11 * 13.So p2, p3, p4 must be 7, 11, 13 in ascending order.Common primes in both products are 7 and 11, which must be p2 and p3.The remaining prime from 385 is 5 (p1), and the remaining prime from 1001 is 13 (p4).Thus the four primes in ascending order are 5, 7, 11, 13.The fourth (largest) prime is 13.

Verification / Alternative check:
Check product of first three: 5 * 7 * 11 = 385, correct. Check product of last three: 7 * 11 * 13 = 1001, also correct. The ordering 5 < 7 < 11 < 13 satisfies the condition of ascending primes.

Why Other Options Are Wrong:
11 and 17: 11 is one of the middle primes, not the largest, and 17 does not appear in the prime factorization of either 385 or 1001.19 and 23: these primes never feature in the factorization of 385 or 1001, so they cannot be any of p1, p2, p3, or p4.

Common Pitfalls:
Incorrect factorization of 385 or 1001, especially confusing 1001 with non-prime factors.Ignoring the ascending order condition and mis-assigning which factors correspond to which positions.Assuming the primes must be consecutive integers, which is not required by the question.

Final Answer:
13