1. **Problem Statement:** Prove that among any 16 consecutive integers, there is at least one integer that is relatively prime (shares no common divisor greater than 1) to each of the others. 2. **Key Idea:** Two numbers are relatively prime if their greatest common divisor (gcd) is 1. 3. **Approach:** Consider 16 consecutive integers: $$n, n+1, n+2, \ldots, n+15$$. 4. **Observation:** If an integer shares a common divisor greater than 1 with any other integer in the set, that divisor must divide their difference. 5. **Differences:** The differences between any two numbers in the set are at most 15. 6. **Prime Divisors:** Any common divisor greater than 1 must be a prime number less than or equal to 15. 7. **Constructing the candidate:** Consider the integer in the set that is congruent to 1 modulo all primes less than or equal to 15. Since the primes less than or equal to 15 are 2, 3, 5, 7, 11, 13, the Chinese Remainder Theorem guarantees an integer in the set congruent to 1 modulo each of these primes. 8. **Conclusion:** This integer cannot share any prime divisor with any other integer in the set because it is 1 modulo those primes, so gcd with any other integer is 1. 9. **Final statement:** Therefore, among any 16 consecutive integers, there exists at least one integer relatively prime to all the others.