Solution: The smallest 4-digit number divisible by 11 is $ 1001 $ (since $ 1001 \div 11 = 91 $), and the largest is $ 9999 $ (since $ 9999 \div 11 = 909 $). The count is $ 909 - 91 + 1 = 819 $.

["The Smallest and Largest 4-Digit Numbers Divisible by 11: A Complete Guide", "When exploring divisibility rules, few numbers spark curiosity as much as those involving 11. One fascinating fact is that the smallest 4-digit number divisible by 11 is 1001, and the largest is 9999. But how do we arrive at these exact values, and how many such numbers exist between them? This article explains everything you need to know about the smallest and largest 4-digit numbers divisible by 11 — including the total count.", "---", "### Understanding Divisibility by 11", "To begin, recall that a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum in even positions is a multiple of 11 (including zero). While this rule helps identify divisibility, it’s equally important to know where 11 divides in the range of 4-digit numbers.", "The smallest 4-digit number is 1000, but 1000 divided by 11 gives a quotient of about 90.909, not an integer. The next candidate divisible by 11 is found by:", "$$\n\lceil 1000 \div 11 \rceil = 91 \quad \Rightarrow \quad 91 \ imes 11 = 1001\n$$", "Thus, 1001 is the smallest 4-digit number divisible by 11.", "For the largest 4-digit number, 9999, direct division yields:", "$$\n9999 \div 11 = 909 \quad \ ext{(exactly)}\n$$", "So, 9999 is not only divisible by 11, but it’s the largest 4-digit number meeting the criteria.", "---", "### Counting All 4-Digit Numbers Divisible by 11", "Now that we’ve identified the first and last terms, we can determine how many such numbers exist. Since all 4-digit multiples of 11 form an arithmetic sequence:", "- First term: $ a = 1001 $\n- Last term: $ l = 9999 $\n- Common difference: $ d = 11 $", "The number of terms in this sequence is calculated using the arithmetic sequence formula:", "$$\nn = \frac{l - a}{d} + 1 = \frac{9999 - 1001}{11} + 1\n$$", "Compute the difference:", "$$\n9999 - 1001 = 8998\n$$", "Now divide:", "$$\n\frac{8998}{11} = 818\n$$", "Add 1 to include both endpoints:", "$$\nn = 818 + 1 = 819\n$$", "---", "### Why This Count Matters", "Understanding how many 4-digit numbers are divisible by 11 has educational, computational, and practical value. This count (819) helps in:", "- Teaching number theory and patterns in arithmetic sequences\n- Optimizing algorithms requiring filtering by divisibility\n- Enhancing logical thinking and problem-solving in math competitions and real-world applications", "---", "### Summary", "- Smallest 4-digit number divisible by 11: 1001\n- Largest 4-digit number divisible by 11: 9999\n- Total count of 4-digit numbers divisible by 11: 819", "So next time you explore divisibility, remember: between 1000 and 9999, exactly 819 four-digit numbers proudly stand as multiples of 11 — starting at 1001, ending at 9999.", "---", "Keywords: smallest 4 digit number divisible by 11, largest 4 digit number divisible by 11, 1001 divisible by 11, 9999 divisible by 11, count of 4 digit multiples of 11, divisibility by 11, mathematics facts, arithmetic sequences, problem solving."]