Question: What is the smallest four-digit number that is divisible by both $12$ and $18$ and leaves a remainder of $3$ when divided by $7$?

Thu, 16 Apr 2026 07:49:42 +0000

**What is the smallest four-digit number that is divisible by both $12$ and $18$ and leaves a remainder of $3$ when divided by $7$?** Curious about how numbers interact in exacting patterns? A question recently circulating among data enthusiasts and problem solvers across the U.S. asks: *What is the smallest four-digit number that is divisible by both 12 and 18 and leaves a remainder of 3 when divided by 7*? Beyond curiosity, this query reflects a broader interest in mathematical relationships, divisibility rules, and modular arithmetic—concepts woven into everyday tech, finance, and problem-solving. With users increasingly seeking precise, verified information on mobile, this query delivers strong relevance in today’s digital landscape. The integer sought is both a multiple of the least common multiple of 12 and 18—and simultaneously satisfies a modular condition. Divisible by both 12 and 18 means the number must be a multiple of their LCM. Since 12 = 2²·3 and 18 = 2·3², their LCM is 2²·3² = 4·9 = 36. So we need the smallest four-digit multiple of 36 that leaves a remainder of 3 when divided by 7. Finding this number begins with identifying the smallest four-digit multiple of 36. The smallest four-digit number is 1000. Dividing 1000 by 36 gives approximately 27.78, so the next whole multiple is 28. Calculating 36 × 28 = 1008—a four-digit start. But we need to check divisibility and modular conditions systematically.