Solution: The number must be divisible by the least common multiple (LCM) of 7, 11, and 13. Since these are all primes, LCM = 7 × 11 × 13 = 1001. The smallest four-digit number is the first multiple of 1001 ≥ 1000, which is $1001 imes 1 = 1001$. Thus, the answer is $oxed{1001}$.

["The Number Must Be Divisible by the LCM of 7, 11, and 13: Find the Smallest Four-Digit Solution", "When tasked with finding the smallest four-digit number divisible by a given set of numbers, understanding the least common multiple (LCM) is key. In this case, the numbers are 7, 11, and 13—each a prime number. Because they share no common factors other than 1, the LCM is simply the product:\nLCM(7, 11, 13) = 7 × 11 × 13 = 1001.", "Now, why is 1001 important? Since we seek the smallest four-digit number that’s divisible by this LCM, we look for the smallest integer multiple of 1001 that meets or exceeds 1000 (the smallest four-digit number).", "The first multiple is:\n1001 × 1 = 1001", "This four-digit number satisfies the condition perfectly—1001 is divisible by 7, 11, and 13, and no smaller four-digit number meets this requirement.", "Why this solution works:\nThe LCM of distinct primes is their product, ensuring strict divisibility. Starting from 1000 and multiplying by 1001 directly gives the threshold that’s both a four-digit number and fully compliant with the divisibility rule.", "Answer: (\boxed{1001})", "For anyone facing similar number theory challenges, identifying the LCM is your first step—coupled with checking the smallest four-digit threshold using that LCM—ensures a fast and accurate solution."]