Question: What two-digit positive integer is one less than a multiple of 5 and also one less than a multiple of 7?
["What Two-Digit Positive Integer Is One Less Than a Multiple of 5 and Also One Less Than a Multiple of 7?", "Finding a two-digit positive integer that is one less than multiples of both 5 and 7 may seem puzzling at first, but it’s actually a clever number puzzle rooted in modular arithmetic. If a number is one less than a multiple of 5 and one less than a multiple of 7, it means the number is congruent to 4 modulo 5 and congruent to 6 modulo 7—but more simply, it’s one less than a common multiple of 5 and 7.", "Let’s break this down step-by-step.", "---", "### Understanding the Problem", "We are looking for a two-digit positive integer ( x ) such that:", "- ( x + 1 ) is divisible by 5\n- ( x + 1 ) is divisible by 7", "In other words:", "[\nx + 1 \equiv 0 \pmod{5} \quad \ ext{and} \quad x + 1 \equiv 0 \pmod{7}\n]", "This means ( x + 1 ) is a common multiple of 5 and 7.", "---", "### Finding the Least Common Multiple", "Since 5 and 7 are coprime (they share no common factors besides 1), their least common multiple (LCM) is simply:", "[\n\ ext{LCM}(5, 7) = 5 \ imes 7 = 35\n]", "So ( x + 1 = 35k ) for some positive integer ( k ).", "Then:", "[\nx = 35k - 1\n]", "---", "### Finding Two-Digit Solutions", "We want ( x ) to be a two-digit number:", "[\n10 \leq 35k - 1 \leq 99\n]", "Add 1 to all parts:", "[\n11 \leq 35k \leq 100\n]", "Now divide by 35:", "[\n\frac{11}{35} \approx 0.314 \leq k \leq \frac{100}{35} \approx 2.857\n]", "So the only integer values ( k ) can take are ( k = 1 ) and ( k = 2 ).", "- For ( k = 1 ): ( x = 35(1) - 1 = 34 )\n- For ( k = 2 ): ( x = 35(2) - 1 = 70 - 1 = 69 )", "Both 34 and 69 are two-digit integers.", "But wait — check if both fit the original condition:", "- ( 34 + 1 = 35 ), divisible by 5 and 7 → ✔\n- ( 69 + 1 = 70 ), divisible by 5 and 7 → ✔", "However, the problem asks for the two-digit positive integer, implying a unique answer. But both 34 and 69 satisfy the condition.", "Let’s re-read: “What two-digit positive integer… is one less than a multiple of 5 and also one less than a multiple of 7?” Since both satisfy, but usually such puzzles expect the smallest or only such number with extra constraints (e.g., only one valid within two-digit range), yet here both do.", "But wait — is there a number that is one less than a multiple of both 5 and 7 and only one such number in the two-digit range? The values are 34 and 69.", "Wait — is there a mistake? Let's verify:", "- ( 34 + 1 = 35 = 5 \ imes 7 ) → yes\n- ( 69 + 1 = 70 = 5 \ imes 14 = 7 \ imes 10 ) → yes", "So both are valid, but the question says “the” integer — perhaps context implies the smallest such number?", "Indeed, in classic puzzles, when two or more exist, the smallest is typically accepted unless restricted.", "Moreover, searching modular equations: ( x \equiv -1 \pmod{5} ) and ( x \equiv -1 \pmod{7} ) implies ( x \equiv 4 \pmod{5} ), ( x \equiv 6 \pmod{7} ), and since 5 and 7 coprime, by Chinese Remainder Theorem, the solution is ( x \equiv -1 \pmod{35} ), i.e., ( x \equiv 34 \pmod{35} ).", "So all solutions are of the form ( x = 35k - 1 ), giving:", "For ( k = 1 ): 34\nFor ( k = 2 ): 69\nFor ( k = 3 ): 104 → three-digit, invalid", "So only two valid two-digit numbers: 34 and 69.", "But since the question uses “the” and expects a single answer, and given educational contexts often expect the smallest such number, we conclude:", "> The smallest two-digit positive integer that is one less than a multiple of both 5 and 7 is 34.", "---", "### Final Answer", "The two-digit positive integer that is one less than a multiple of 5 and one less than a multiple of 7 is:", "34", "*(Note: 69 also satisfies the condition; however, 34 is the smallest and most commonly"]
