Next, find numbers divisible by both 7 and 5 (i.e., 35): Smallest 3-digit is 105 ($ 35 \times 3 $), largest 980 ($ 35 \times 28 $). Total: $ 28 - 2 = 26 $.
["Finding All 3-Digit Numbers Divisible by Both 5 and 7 (i.e., 35)", "When tasked with finding three-digit numbers divisible by both 5 and 7, you’re essentially looking for numbers divisible by their least common multiple (LCM). Since 5 and 7 are coprime, their LCM is simply:", "[\n\ ext{LCM}(5, 7) = 5 \ imes 7 = 35\n]", "So, identifying numbers divisible by both 5 and 7 means finding all 3-digit multiples of 35.", "### Step-by-Step Guide to Identify the Range", "We want all 3-digit numbers from 100 to 999 inclusive that are divisible by 35.", "Start by finding the smallest 3-digit number divisible by 35:", "- The smallest 3-digit number is 100\n- Divide 100 by 35:\n [\n 100 \div 35 \approx 2.857\n ]\n The next whole number is 3.\n So,\n [\n 35 \ imes 3 = 105\n ]\n105 is the smallest 3-digit number divisible by 35.", "Now, find the largest 3-digit number divisible by 35:", "- The largest 3-digit number is 999\n- Divide 999 by 35:\n [\n 999 \div 35 \approx 28.541\n ]\n Take the integer part: 28\n So,\n [\n 35 \ imes 28 = 980\n ]\n980 is the largest 3-digit number divisible by 35.", "### Counting the Multiples", "The sequence of 3-digit multiples of 35 is:", "[\n35 \ imes 3 = 105,\ 35 \ imes 4 = 140,\ \dots,\ 35 \ imes 28 = 980\n]", "This forms an arithmetic sequence where:", "- First term ((a_1)) = 3\n- Last term ((a_n)) = 28\n- Common difference = 1 (each next multiple corresponds to the next integer)", "The number of terms is:\n[\nn = 28 - 3 + 1 = 26\n]", "### Summary", "- Smallest 3-digit number divisible by both 5 and 7: 105 ($35 \ imes 3$)\n- Largest: 980 ($35 \ imes 28$)\n- Total count: 26 numbers", "This means there are 26 three-digit numbers divisible by 35 — all multiples of 35 between 105 and 980 inclusive.", "Understanding multiples of 35 makes finding numbers divisible by both 5 and 7 straightforward and efficient. Whether you’re solving math problems, coding algorithms, or analyzing number patterns, recognizing this relationship helps streamline calculations.", "---", "Keywords: numbers divisible by 7 and 5, 35 multiple three-digit numbers, smallest 3-digit multiple of 35, largest 3-digit multiple of 35, count of 35 multiples between 100 and 999, 35 × 3 to 35 × 28, divisible by both 5 and 7, find numbers divisible by 35"]
