Thus, the number of positive 3-digit numbers divisible by 7 is \(\boxed{128}\).

["# Thus, the Number of Positive 3-Digit Numbers Divisible by 7 Is (\boxed{128})", "If you're wondering how many positive 3-digit numbers are divisible by 7, the answer is precisely 128. This straightforward yet insightful number reveals a clean mathematical pattern and offers useful practice in divisibility and arithmetic sequences.", "## What Are 3-Digit Numbers?", "Three-digit numbers range from 100 to 999, inclusive. To find how many of these are divisible by 7, we determine the first and last such numbers in the range and count how many lie between them with a step size of 7.", "## Step-by-Step Reasoning", "1. Find the smallest 3-digit number divisible by 7\n Divide 100 by 7:\n [\n \frac{100}{7} \approx 14.2857\n ]\n The next whole number is 15. Multiply:\n [\n 15 \ imes 7 = 105\n ]\n So, 105 is the smallest 3-digit number divisible by 7.", "2. Find the largest 3-digit number divisible by 7\n Divide 999 by 7:\n [\n \frac{999}{7} \approx 142.714\n ]\n The largest whole number less than or equal to this is 142.\n [\n 142 \ imes 7 = 994\n ]\n So, 994 is the largest 3-digit number divisible by 7.", "3. Count the total numbers in this sequence\n We now have an arithmetic sequence starting at 105, ending at 994, with a common difference of 7.\n Use the formula for the number of terms in an arithmetic sequence:\n [\n \ ext{Number of terms} = \frac{\ ext{Last term} - \ ext{First term}}{\ ext{Common difference}} + 1\n ]\n Substituting the values:\n [\n \frac{994 - 105}{7} + 1 = \frac{889}{7} + 1 = 127 + 1 = 128\n ]", "## Why Is the Count Exactly 128?", "Each positive 3-digit number divisible by 7 falls perfectly into the sequence:\n105, 112, 119, ..., 994\nThis sequence increments by 7, systematically covering every multiple of 7 between 100 and 999. Since 994 − 105 = 889, and dividing this by 7 gives 127 intervals, adding 1 gives exactly 128 numbers.", "## Practical Applications and Why This Matters", "Understanding how many multiples of 7 lie within a range helps in scheduling, resource allocation, data sampling, and more. This concept is vital in modular arithmetic, combinatorics, and algorithmic thinking.", "## Conclusion", "The number of positive 3-digit numbers divisible by 7 is exactly (\boxed{128}). This clear and computable result highlights the elegance of number theory and serves as a practical example in teaching division, sequences, and divisibility.", "If you're practicing math puzzles or studying divisibility, knowing such counts reinforces both calculation skills and conceptual understanding. Next time you explore 3-digit numbers, remember: the answer is 128, neatly boxed and mathematically proven."]