Question: How many positive 3-digit numbers are divisible by 7 but not by 5?
["How Many Positive 3-Digit Numbers Are Divisible by 7 but Not by 5?", "When exploring numbers within specific mathematical sets, one intriguing question is: How many positive 3-digit numbers are divisible by 7 but not by 5? This seemingly simple query opens the door to understanding patterns in number theory, divisibility rules, and counting techniques. In this article, we’ll break down the solution step by step, offering clear insights and practical calculations anyone can follow.", "---", "### Understanding the Range of 3-Digit Numbers", "A 3-digit number ranges from 100 to 999, inclusive. We are interested in those numbers within this interval that satisfy two conditions:", "1. Divisible by 7\n2. Not divisible by 5", "---", "### Step 1: Count How Many 3-Digit Numbers Are Divisible by 7", "To find all 3-digit numbers divisible by 7, we:", "- Identify the smallest 3-digit number divisible by 7\n- Identify the largest 3-digit number divisible by 7\n- Count all multiples in between", "Smallest:\nThe smallest 3-digit number is 100.\nDivide 100 by 7:\n[ 100 ÷ 7 ≈ 14.285 ]\nThe next whole number is 15, so:\n[ 15 × 7 = 105 ]\nThus, 105 is the smallest 3-digit number divisible by 7.", "Largest:\nThe largest 3-digit number is 999.\nDivide 999 by 7:\n[ 999 ÷ 7 ≈ 142.714 ]\nTake the floor:\n[ 142 × 7 = 994 ]\nSo, 994 is the largest 3-digit number divisible by 7.", "Now count the multiples:\n[ \ ext{Count} = \left( \frac{994 - 105}{7} \right) + 1 = \left( \frac{889}{7} \right) + 1 = 127 + 1 = 128 ]", "✅ There are 128 three-digit numbers divisible by 7.", "---", "### Step 2: Exclude Numbers Also Divisible by 5", "A number divisible by both 7 and 5 is divisible by their least common multiple, 35. So, we need to subtract the 3-digit numbers divisible by 35 from the 128 counted above.", "Find how many 3-digit numbers are divisible by 35:", "Smallest 3-digit multiple of 35:\n[ 100 ÷ 35 ≈ 2.857 \Rightarrow \lceil 2.857 \rceil = 3 ]\n[ 3 × 35 = 105 ]", "Largest 3-digit multiple of 35:\n[ 999 ÷ 35 ≈ 28.542 \Rightarrow \lfloor 28.542 \rfloor = 28 ]\n[ 28 × 35 = 980 ]", "Count of multiples:\n[ \frac{980 - 105}{35} + 1 = \frac{875}{35} + 1 = 25 + 1 = 26 ]", "✅ So, 26 three-digit numbers are divisible by both 7 and 5.", "---", "### Step 3: Compute the Final Answer", "To find 3-digit numbers divisible by 7 but not by 5, subtract the overlap from the total:", "[ 128 \ ext{ (div by 7)} - 26 \ ext{ (div by both 7 and 5)} = 102 ]", "---", "### Conclusion", "There are 102 positive 3-digit numbers divisible by 7 but not by 5. This calculation combines basic divisibility principles, efficient counting within ranges, and clear subtraction to exclude unwanted cases—excellent practice for mastering number theory and real-world problem-solving.", "Whether you're solving math problems, building algorithms, or just curious, understanding how to filter numbers by divisibility criteria sharpens essential logical skills!", "---", "### Key Takeaways:", "- The smallest 3-digit multiple of 7 is 105\n- The largest is 994 → 128 numbers total\n- The least common multiple of 7 and 5 is 35\n- There are 26 numbers divisible by 35 in the 3-digit range\n- Final count: 128 – 26 = 102", "---", "Keywords: 3-digit numbers divisible by 7, not divisible by 5, count 7 divisible by 35, math problem solving, divisibility rules, number theory, counting multiples of 7 and 5, 3-digit number calculation.", "---", "Explore more mathematical curiosities and find precise answers with logic and simple arithmetic!"]
