Thus, the remainder is $ \boxed{2} $.Question: What is the smallest four-digit number that is divisible by the least common multiple of 12 and 18, and also leaves a remainder of 3 when divided by 7?

["Finding the Smallest Four-Digit Number: Divisible by LCM of 12 and 18, and Leaves Remainder 3 When Divided by 7", "When tackling number puzzles involving divisibility and remainders, precision and methodical reasoning are key. One intriguing problem asks for the smallest four-digit number divisible by the least common multiple (LCM) of 12 and 18—and also leaves a remainder of 3 when divided by 7. Let’s break this down step by step.", "Step 1: Compute the LCM of 12 and 18", "To find the smallest number divisible by both 12 and 18, we calculate their LCM.\n- Prime factorization:\n - $12 = 2^2 \ imes 3$\n - $18 = 2 \ imes 3^2$\n- The LCM takes the highest powers:\n - $ \ ext{LCM}(12, 18) = 2^2 \ imes 3^2 = 4 \ imes 9 = 36 $", "So, we seek the smallest four-digit number divisible by 36 and such that:\n[ x \equiv 3 \pmod{7} ]", "Step 2: Find the smallest four-digit multiple of 36", "The smallest four-digit number is 1000.\nDivide 1000 by 36:\n[ 1000 \div 36 \approx 27.78 ]\nSo, the smallest integer greater than or equal to 27.78 is 28.\nThus, the smallest four-digit multiple of 36 is:\n[ 36 \ imes 28 = 1008 ]", "We now check if 1008 satisfies the remainder condition:\n[ 1008 \div 7 = 144 \ ext{ remainder } 0 ]\nBut we need remainder 3:\n[ 1008 \equiv 0 \pmod{7},\ \ ext{not } 3 ]", "So, 1008 is too small. We must find the next four-digit multiples of 36 and check which satisfies $x \equiv 3 \pmod{7}$.", "We proceed sequentially through multiples of 36 starting from 1008:\n- $1008 + 36 = 1044$\n- $1044 + 36 = 1080$\n- $1080 + 36 = 1116$\n- $1116 + 36 = 1152$\n- $1152 + 36 = 1188$\n- $1188 + 36 = 1224$\n- $1224 + 36 = 1260$\n- $1260 + 36 = 1296$\n- $1296 + 36 = 1332$\n- $1332 + 36 = 1368$", "Now check each modulo 7:", "- $1008 \mod 7 = 0$\n- $1044 \mod 7$: $1044 \div 7 = 149 \ imes 7 = 1043$, remainder $1$\n- $1080 \div 7$: $7 \ imes 154 = 1078$, remainder $2$\n- $1116 \div 7$: $7 \ imes 159 = 1113$, remainder $3$ → Match found!", "Step 3: Verify the solution", "- Is 1116 a four-digit number? ✅\n- Is $1116 \div 36 = 31$ → whole number? $36 \ imes 31 = 1116$ ✅\n- Does $1116 \equiv 3 \pmod{7}$? $1116 - 3 = 1113$, and $1113 \div 7 = 159$, an integer ✅", "Therefore, 1116 satisfies all conditions: divisible by 36 (LCM of 12 and 18), and leaves a remainder of 3 when divided by 7.", "Conclusion:\nThus, the smallest four-digit number satisfying both criteria is:\n[ \boxed{1116} ]", "This problem illustrates how combining LCM with modular arithmetic helps efficiently solve multi-condition number puzzles—essential for math enthusiasts, developers solving algorithmic challenges, and students sharpening logical reasoning in number theory."]