\boxed{9}Question: What is the smallest positive integer that is a multiple of 16 and one more than a multiple of 5?
["The Smallest Positive Integer That Is a Multiple of 16 and One More Than a Multiple of 5", "Finding numbers that satisfy multiple conditions can be intriguing, especially when exploring integer mathematics. One fascinating problem is: What is the smallest positive integer that is a multiple of 16 and one more than a multiple of 5? This question combines divisibility rules with modular arithmetic, leading to a clear and elegant solution.", "### Understanding the Problem", "We are looking for the smallest positive integer ( x ) that meets two conditions:\n1. ( x ) is a multiple of 16 → ( x \equiv 0 \pmod{16} )\n2. ( x ) is one more than a multiple of 5 → ( x \equiv 1 \pmod{5} )", "In mathematical terms:\n[\n\begin{align}\nx &\equiv 0 \pmod{16} \\nx &\equiv 1 \pmod{5}\n\end{align}\n]", "### Solving the System of Congruences", "This is a classic system of linear congruences that can be solved using the Chinese Remainder Theorem or by direct search, since modulus 16 and 5 are small.", "We seek integers ( x ) such that:\n- ( x = 16k ) for some integer ( k )\n- Substituting into the second condition:\n[\n16k \equiv 1 \pmod{5}\n]", "Now simplify modulo 5:\n[\n16 \equiv 1 \pmod{5} \quad \ ext{since } 16 \div 5 = 3 \ ext{ remainder } 1\n]\nSo the congruence becomes:\n[\n1 \cdot k \equiv 1 \pmod{5} \implies k \equiv 1 \pmod{5}\n]", "Thus, ( k = 5m + 1 ) for some integer ( m ).\nSubstitute back into ( x = 16k ):\n[\nx = 16(5m + 1) = 80m + 16\n]", "The general solution is ( x = 80m + 16 ). The smallest positive value occurs when ( m = 0 ):\n[\nx = 80(0) + 16 = 16\n]", "### Verifying the Solution", "Check whether 16 satisfies both conditions:\n- Is 16 a multiple of 16? Yes, ( 16 \div 16 = 1 ).\n- Is 16 one more than a multiple of 5?\n[\n16 - 1 = 15, \quad \ ext{and } 15 \div 5 = 3 \ ext{ — an integer.}\n]\nSo, 16 satisfies both conditions.", "### Conclusion", "The smallest positive integer that is both a multiple of 16 and one more than a multiple of 5 is:", "[\n\boxed{16}\n]", "This elegant solution combines number theory fundamentals and modular logic, making it a classic example of how mathematical reasoning can uncover precise answers to seemingly complex problems."]
