Question: What two-digit positive integer is one less than a multiple of 9 and also one more than a multiple of 7?
["What Two-Digit Positive Integer Is One Less Than a Multiple of 9 and One More Than a Multiple of 7?", "Are you curious about a special two-digit number that meets a very specific mathematical condition? If so, you’re in the right place! The answer to the question: What two-digit positive integer is one less than a multiple of 9 and one more than a multiple of 7? is revealed with a simple yet elegant solution rooted in modular arithmetic.", "### Decoding the Clues", "We’re looking for a two-digit integer, let’s call it x, such that:", "1. x is one less than a multiple of 9 → ( x \equiv -1 \pmod{9} )\n This is equivalent to:\n [\n x \equiv 8 \pmod{9}\n ]", "2. x is one more than a multiple of 7 → ( x \equiv 1 \pmod{7} )", "Now we need to solve this system of congruences:\n[\n\begin{cases}\nx \equiv 8 \pmod{9} \\nx \equiv 1 \pmod{7}\n\end{cases}\n]", "### Solving the System Using the Chinese Remainder Theorem (CRT)", "Since 9 and 7 are coprime (gcd(9,7)=1), the Chinese Remainder Theorem guarantees a unique solution modulo ( 9 \ imes 7 = 63 ).", "We begin by expressing ( x ) from the first congruence:\n[\nx = 9k + 8 \quad \ ext{for some integer } k\n]", "Substitute this into the second congruence:\n[\n9k + 8 \equiv 1 \pmod{7}\n]", "Simplify:\n[\n9k \equiv 1 - 8 \equiv -7 \equiv 0 \pmod{7}\n]\nSince ( 9 \equiv 2 \pmod{7} ), this becomes:\n[\n2k \equiv 0 \pmod{7}\n]", "Dividing both sides by 2 (or multiplying by the modular inverse of 2 modulo 7, which is 4, because ( 2 \cdot 4 = 8 \equiv 1 \pmod{7} )):\n[\nk \equiv 0 \cdot 4 = 0 \pmod{7}\n]", "So:\n[\nk = 7m \quad \ ext{for some integer } m\n]", "Substitute back into ( x = 9k + 8 ):\n[\nx = 9(7m) + 8 = 63m + 8\n]", "Thus, solutions are of the form:\n[\nx \equiv 8 \pmod{63}\n]", "### Finding the Two-Digit Solution", "We now find values of ( m ) such that ( x = 63m + 8 ) is a two-digit number (i.e., between 10 and 99).\nTry ( m = 1 ):\n[\nx = 63(1) + 8 = 71\n]\nTry ( m = 0 ):\n[\nx = 63(0) + 8 = 8 \quad \ ext{(not two-digit)}\n]\nTry ( m = 2 ):\n[\nx = 63(2) + 8 = 126 + 8 = 134 \quad \ ext{(too large)}\n]", "The only two-digit solution is:\n[\nx = 71\n]", "### Verifying the Answer", "Check both conditions:", "- Is 71 one less than a multiple of 9?\n ( 71 + 1 = 72 ), and ( 72 \div 9 = 8 ) → ✅\n- Is 71 one more than a multiple of 7?\n ( 71 - 1 = 70 ), and ( 70 \div 7 = 10 ) → ✅", "Both conditions are satisfied.", "### Conclusion", "The two-digit positive integer that is one less than a multiple of 9 and one more than a multiple of 7 is:\n71", "This elegant problem highlights how modular arithmetic can uncover hidden patterns in numbers—perfect for math enthusiasts, students, and anyone interested in number theory.", "[\n\boxed{71}\n]"]
