Thus, there are \(\boxed{81}\) 3-digit numbers divisible by 11.

["# How Many Are There? Exploring the Count of 3-Digit Numbers Divisible by 11", "Understanding numbers and their properties is essential in mathematics, and one intriguing fact is exactly how many 3-digit numbers are divisible by 11. Careful examination reveals there are precisely (\boxed{81}) such numbers. This number isn’t arbitrary—it reflects patterns in divisibility and the structure of three-digit ranges. Let’s explore how we arrive at this result.", "## What Makes a Number Divisible by 11?", "A number is divisible by 11 if it satisfies the divisibility rule: the difference between the sum of digits in odd positions and the sum of digits in even positions must be a multiple of 11 (including zero). For 3-digit numbers, represented as (ABC) (where (A), (B), and (C) are digits and (A <br/>\neq 0)), this means:\n[\n(A + C) - B \equiv 0 \pmod{11}\n]\nHowever, counting divisible numbers efficiently doesn’t require checking each integer—there’s a mathematical shortcut.", "## Identifying the Range of 3-Digit Numbers", "Three-digit numbers start from 100 and end at 999. To find how many lie within this set and are divisible by 11, we determine the smallest and largest 3-digit numbers divisible by 11.", "- The smallest 3-digit number divisible by 11 is (110), since (100 \div 11 \approx 9.09), so the next whole multiple is (11 \ imes 10 = 110).\n- The largest is (990), because (999 \div 11 \approx 90.818), and the floor multiple is (11 \ imes 90 = 990).", "## Counting Using Arithmetic Sequences", "The numbers divisible by 11 between 100 and 999 form an arithmetic sequence where:\n- First term ((a_1)) = 110\n- Common difference ((d)) = 11\n- Last term ((a_n)) = 990", "To find the total count of terms ((n)), use the formula for the (n)-th term of an arithmetic sequence:\n[\na_n = a_1 + (n-1) \ imes d\n]\nPlugging in known values:\n[\n990 = 110 + (n-1) \ imes 11\n]\nSubtract 110:\n[\n880 = (n-1) \ imes 11\n]\nDivide by 11:\n[\nn - 1 = 80\n]\nThus:\n[\nn = 81\n]", "## Why There Are Exactly 81 Such Numbers", "This count arises because nodes (divisible numbers) evenly space every 11 units across naturally occurring integers. Since 110 to 990 includes exactly 81 multiples of 11 without gaps (due to correct initial and final terms), every valid 3-digit number divisible by 11 falls uniquely into this sequence.", "## Why This Matters in Education and Logic", "Understanding how to calculate counts like (\boxed{81}) strengthens mathematical reasoning. It illustrates how patterns in modular arithmetic simplify counting tasks and reinforces fluency in working with sequences. Teachers, students, and math enthusiasts alike benefit from recognizing such structures—they make numbers less mysterious and more logical.", "## Conclusion", "There are exactly (\boxed{81}) three-digit numbers divisible by 11, a result born of precise arithmetic sequences and divisibility rules. Whether for calculative ease or conceptual clarity, mastering such facts enriches problem-solving skills and deepens numerical intuition. Start counting, and uncover the elegant order hiding behind seemingly complex sets!"]