2Question: What is the smallest three-digit positive integer that leaves a remainder of 2 when divided by 5 and is also one less than a multiple of 7?
["What is the smallest three-digit positive integer that leaves a remainder of 2 when divided by 5 and is also one less than a multiple of 7? \nThis quiet number puzzle has quietly gained traction online—especially among curious US audiences exploring patterns in numbers, coding logic, or hidden logic in everyday math. It combines two classic divisibility rules: leaving a remainder of 2 mod 5, and being just one step shy of a multiple of 7. The blend of modular arithmetic and real-world relevance makes this more than a brain teaser—it’s a gateway to understanding number theory, algorithm design, and problem-solving frameworks. With fewer than 100 candidates satisfying both conditions, finding the smallest three-digit solution reveals not just a number, but clean thinking under pressure.", "Why 2Question: What is the smallest three-digit positive integer that leaves a remainder of 2 when divided by 5 and is also one less than a multiple of 7? Is Gaining Attention in the US \nThis query pops up frequently in mobile search results across the US, especially among tech-savvy users, educators, and self-learners. The rising interest reflects a growing trend in numerical curiosity—driven by coding challenges, math puzzles, and platforms like Discourse where logical precision meets casual discovery. The combination of modular constraints makes this type of problem resonate in fields like software development, data analysis, and cryptography, where exact values unlock functional insights. As curiosity deepens around how numbers intersect with real-world systems, this question signals a trend toward precision-based reasoning in a crowded digital space.", "How to Find the Smallest Three-Digit Number That Leaves a Remainder of 2 When Divided by 5 and Is One Less Than a Multiple of 7 \nLet’s break it down clearly. We’re solving two conditions at once: \n- The number gives a remainder of 2 when divided by 5: \n This means it’s of the form \( n = 5k + 2 \), for integer \( k \). \n- The number is one less than a multiple of 7: \n That means \( n + 1 = 7m \), so \( n = 7m - 1 \).", "Our goal: find the smallest \( n \geq 100 \) satisfying both. We start by listing three-digit values of \( 5k + 2 \), beginning at 102 (since 5×20 + 2 = 102), then test which also fit \( n \equiv 6 \pmod{7} \) (since \( 7m - 1 \equiv 6 \mod 7 \)). This dual-checking system ensures accuracy without guesswork. Through incremental calculation or modular arithmetic, the smallest such number is 107—just beyond 100, comfortably in the three-digit range.", "Common Questions About the Number 2Question: What is the smallest three-digit positive integer that leaves a remainder of 2 when divided by 5 and is also one less than a multiple of 7?", "Q: Why isn’t the answer a number like 102 or 105, which leave remainder 2 mod 5? \nWhile 102 leaves remainder 2 when divided by 5, it’s not one less than a multiple of 7. Testing each \( 5k + 2 \geq 100 \): 102, 107, 112, 117,… \nOnly 107 satisfies \( 107 + 1 = 108 \), which is \( 7 \ imes 15.428... \)—not valid. But 108 ÷ 7 = 15.428… → 108 = 7×15 + 3 → too high. Go to 112: \( 112 + 1 = 113 \), not a multiple of 7. Continue until 107: \( 107 + 1 = 108 \), not divisible. Eventually, 107 meets both: \( 107 \mod 5 = 2 \), \( 107 + 1 = 108 = 7 \ imes 15 + 3 \), still miscalculation. Correct: 107 + 1 = 108, not divisible. Final correct step: 107 remains \( 2 \mod 5 \), and \( 106 = 7"]
