Question: What is the smallest three-digit number that is divisible by both $12$ and $15$ and leaves a remainder of $1$ when divided by $7$?
["Mastering Number Theory: The Smallest Three-Digit Number Divisible by 12, 15, and Leaves Remainder 1 When Divided by 7", "When exploring the fascinating world of number theory, some problems challenge our understanding of divisibility, modular arithmetic, and constraints. One such intriguing question is:", "What is the smallest three-digit number that is divisible by both 12 and 15 and leaves a remainder of 1 when divided by 7?", "This article guides you step-by-step through solving this complex yet rewarding problem—uncovering the solution while illuminating core mathematical concepts.", "---", "## Understanding the Divisibility Condition", "First, let’s break down the divisibility requirement.", "A number divisible by both 12 and 15 must be divisible by the least common multiple (LCM) of these two numbers.", "- Prime factorization:\n - $12 = 2^2 \cdot 3$\n - $15 = 3 \cdot 5$", "The LCM takes the highest powers of all primes involved:\n$$\n\ ext{LCM}(12, 15) = 2^2 \cdot 3 \cdot 5 = 60\n$$", "So, we are looking for the smallest three-digit multiple of 60 that satisfies an additional modular condition.", "---", "## Applying the Modular Arithmetic Condition", "The number must also satisfy:\n$$\nn \equiv 1 \pmod{7}\n$$\nThis means when $n$ is divided by 7, the remainder is 1.", "---", "## Finding the Smallest Three-Digit Multiple of 60", "Three-digit numbers range from 100 to 999. The smallest three-digit multiple of 60 is:\n$$\n60 \ imes 2 = 120 \quad (\ ext{but } 120 < 100 \ ext{ is acceptable since it's the first three-digit one})\n$$", "But note: 60 × 1 = 60 (too small), so we begin checking multiples from:\n$$\n60 \ imes 2 = 120,\ 60 \ imes 3 = 180,\ 60 \ imes 4 = 240,\ \dots\n$$", "We list the multiples of 60 until we find one satisfying $n \equiv 1 \pmod{7}$:", "- $60 \ imes 2 = 120$ → $120 \div 7 = 17 \ imes 7 = 119$, remainder $120 - 119 = 1$ → Remainder = 1", "✅ 120 satisfies both conditions!", "---", "## Verifying All Conditions", "1. Three-digit? Yes — 120 is between 100 and 999.\n2. Divisible by 12? $120 \div 12 = 10$ → Yes\n3. Divisible by 15? $120 \div 15 = 8$ → Yes\n4. Leaves remainder 1 when divided by 7? $120 \div 7 = 17$ with remainder $1$ → Yes", "All conditions are satisfied.", "---", "## Why Is 120 the Answer?", "Because 60 is the smallest number divisible by both 12 and 15, and it happens to leave a remainder of 1 modulo 7. Any smaller three-digit number would either not meet the divisibility by 60 or fail the modulo condition.", "---", "## Alternative Explanation: Using Modular Arithmetic (for deeper insight)", "Let $n = 60k$, where $k$ is a positive integer and $n \geq 100$.\nWe want:\n$$\n60k \equiv 1 \pmod{7}\n$$", "First, reduce 60 modulo 7:\n$$\n60 \div 7 = 8 \ imes 7 = 56,\quad 60 - 56 = 4 \Rightarrow 60 \equiv 4 \pmod{7}\n$$", "So:\n$$\n4k \equiv 1 \pmod{7}\n$$", "Now solve for $k$: find the multiplicative inverse of 4 modulo 7.", "Try small values:\n- $4 \ imes 1 = 4 \equiv 4$\n- $4 \ imes 2 = 8 \equiv 1 \pmod{7}$ → Inverse is 2", "Thus:\n$$\nk \equiv 2 \pmod{7}\n$$", "So $k = 7m + 2$, for integer $m \geq 0$", "Now compute smallest $k$ such that $60k \geq 100$:", "Try $m = 0$: $k = 2$ → $n = 60 \ imes 2 = 120$ → valid", "Next solution: $k = 9$ → $n = 540$, which is larger than 120", "Thus, the smallest such number is 120", "---", "## Conclusion: The Answer and Its Significance", "The smallest three-digit number divisible by both 12 and 15 and leaving a remainder of 1 when divided by 7 is\n120.", "This problem elegantly combines LCM for compound divisibility, modular arithmetic, and number constraints—essential tools in mathematical problem-solving.", "Whether you’re a student mastering number theory or a enthusiast exploring mathematical puzzles, solving such problems strengthens logical thinking and deepens conceptual understanding.", "---", "## Tips for Similar Problems:", "- Always compute LCM for multiple divisibility requirements.\n- Express the number in terms of the LCM and apply the modulo condition.\n- Use modular inverses when solving equations like $ax \equiv b \pmod{m}$.\n- Check the smallest candidate first, then increment using the found congruence pattern.", "---", "Keywords: smallest three-digit number divisible by 12 and 15, remainder 1 when divided by 7, LCM 60, modular arithmetic, solve number theory problem, divisibility conditions, mathematical puzzle.", "Meta Description:\nDiscover how to find the smallest three-digit number divisible by both 12 and 15 and leaves a remainder of 1 modulo 7. Learn step-by-step via LCM, modular arithmetic, and verification. Perfect for math learners and number theory enthusiasts."]
