Four prime numbers are arranged in increasing order. The product of the first three primes is 7429, while the product of the last three primes is 12673. What is the value of the fourth prime number?

Explanation

Let the four primes be p1 < p2 < p3 < p4. Given p1 × p2 × p3 = 7429 and p2 × p3 × p4 = 12673. Dividing the second product by the first gives (p2 × p3 × p4) / (p1 × p2 × p3) = p4 / p1 = 12673 / 7429 = 29 / 17. Hence, p4 = 29 and p1 = 17. Therefore, the fourth prime number is 29.