Solution: First, find 3-digit numbers divisible by 7: The smallest is 105 ($ 7 \times 15 $), the largest is 994 ($ 7 \times 142 $). Total numbers: $ 142 - 14 = 128 $.
["Solution: Finding All 3-Digit Numbers Divisible by 7", "When tasked with identifying 3-digit numbers divisible by 7, understanding how to systematically determine the range and count efficiently makes all the difference. This informative guide breaks down the complete solution, from locating the smallest and largest 3-digit multiples of 7 to calculating the total count with confidence.", "---", "### Step 1: Identifying the Smallest 3-Digit Number Divisible by 7", "A 3-digit number ranges from 100 to 999. To find the smallest such number divisible by 7, divide the lower bound by 7:", "[\n\left\lceil \frac{100}{7} \right\rceil = \left\lceil 14.2857 \right\rceil = 15\n]", "This means 7 multiplied by 15 gives the first 3-digit multiple of 7:\n[\n7 \ imes 15 = 105\n]\nSo, 105 is the smallest 3-digit number divisible by 7.", "---", "### Step 2: Locating the Largest 3-Digit Number Divisible by 7", "Now, find the largest 3-digit number divisible by 7, capping at 999:\n[\n\left\lfloor \frac{999}{7} \right\rfloor = \left\lfloor 142.714 \right\rfloor = 142\n]", "Thus, the largest multiple is:\n[\n7 \ imes 142 = 994\n]\nTherefore, 994 is the highest 3-digit number divisible by 7.", "---", "### Step 3: Counting All 3-Digit Multiples of 7", "With the smallest (105) and largest (994) values identified, and knowing multiples of 7 form an arithmetic sequence:\n- First term: 105\n- Last term: 994\n- Common difference: 7", "The total count is computed using the formula:\n[\n\ ext{Count} = (\ ext{Last} – \ ext{First}) \div d + 1\n]\nSubstituting the known values:\n[\n\ ext{Count} = (994 - 105) \div 7 + 1 = 889 \div 7 + 1 = 127 + 1 = 128\n]", "Hence, there are 128 three-digit numbers divisible by 7.", "---", "### Why This Method Works", "By transforming the problem into finding endpoints within the sequence of multiples of 7, we avoid tedious enumeration. The ceiling and floor functions adjust the range precisely, while arithmetic progression formulas provide a fast, accurate count.", "---", "### Practical Applications", "This method applies broadly—not just to 7, but to any divisor. Whether teaching math students or solving logical puzzles, recognizing 3-digit multiples of a number saves time and ensures correctness.", "---", "### Conclusion", "Knowing the smallest (105), largest (994), and total count (128) of 3-digit numbers divisible by 7 is simple using arithmetic reasoning and basic division:\n- Smallest: $ 7 \ imes 15 = 105 $\n- Largest: $ 7 \ imes 142 = 994 $\n- Total: $ 142 - 14 = 128 $", "Use this solution to quickly solve similar divisibility problems with confidence.", "---", "Keywords: 3-digit numbers divisible by 7, find multiples of 7, smallest 3-digit multiple of 7, largest 3-digit multiple of 7, total 3-digit multiples of 7, arithmetic sequence formula, math problem solution, divisibility math, easy counting method."]
