There are \(\boxed{1800}\) positive 4-digit numbers divisible by 5.
["Understanding Why There Are Exactly 1,800 Positive 4-Digit Numbers Divisible by 5", "When analyzing how many 4-digit numbers are divisible by 5, mathematics reveals a straightforward yet fascinating pattern. The total number of positive 4-digit numbers runs from 1000 to 9999, inclusive. To find how many of these are divisible by 5, we use basic arithmetic and number theory.", "### What Makes a Number Divisible by 5?", "A number is divisible by 5 if and only if its last digit is either 0 or 5. This binary condition simplifies counting across large sequences.", "### The Range of 4-Digit Numbers", "- Smallest 4-digit number: 1000\n- Largest 4-digit number: 9999", "We now identify the smallest and largest 4-digit numbers divisible by 5:", "- The smallest multiple of 5 ≥ 1000:\n Since 1000 ÷ 5 = 200, and 5 × 200 = 1000 → 1000 is divisible by 5\n- The largest multiple of 5 ≤ 9999:\n Since 9999 ÷ 5 = 1999.8 → Round down to nearest integer: 1999\n 5 × 1999 = 9995", "Thus, the sequence of 4-digit numbers divisible by 5 is:\n1000, 1005, 1010, ..., 9995\nThis is an arithmetic sequence where:\n- First term ((a_1)) = 1000\n- Common difference ((d)) = 5\n- Last term ((a_n)) = 9995", "### Finding the Number of Terms", "The formula for the (n)-th term of an arithmetic sequence is:\n[\na_n = a_1 + (n - 1)d\n]", "Plugging in known values:\n[\n9995 = 1000 + (n - 1) \cdot 5\n]", "Subtract 1000:\n[\n8995 = (n - 1) \cdot 5\n]", "Divide both sides by 5:\n[\n1799 = n - 1\n]", "Add 1:\n[\nn = 1800\n]", "### Summary", "- Total 4-digit numbers: 9000 (from 1000 to 9999)\n- Exactly 1800 of them are divisible by 5\n- This result follows from the predictable arithmetic pattern and simple division", "Understanding this pattern helps in quickly solving similar problems involving divisibility and counting within number ranges. Whether you're teaching math, solving coding challenges, or analyzing data, knowing how many numbers satisfy specific divisibility conditions saves time and improves accuracy.", "So there are exactly (\boxed{1800}) positive 4-digit numbers divisible by 5 — a clean, logical total rooted in arithmetic sequence principles."]
