Better approach: we are to find the smallest $ B $ divisible by 11 such that $ B $ divides some $ D \in [100,199] $ with $ 9 \mid D $.

["Optimizing the Search for the Smallest Multiple of 11: Finding the Minimal $ B $ within Range That Divides a 3-Digit Number Divisible by 9", "In number theory problems involving divisibility, identifying the smallest number $ B $ satisfying multiple constraints efficiently is both a practical task and a demonstration of logical reasoning. This article explores a precise and effective approach to find the smallest $ B $, divisible by 11, such that $ B $ divides at least one integer $ D $ in the range $ [100, 199] $ where $ 9 \mid D $ (i.e., $ D $ is divisible by 9).", "---", "### Problem Restatement", "Find the smallest integer $ B $ satisfying:", "1. $ B $ is divisible by 11 → $ B \equiv 0 \pmod{11} $,\n2. $ 100 \leq B \leq 199 $,\n3. There exists at least one multiple $ D \in [100, 199] $ such that $ 9 \mid D $ and $ B \mid D $.", "---", "### Step-by-Step Strategy", "#### Step 1: List Multiples of 11 in [100, 199]", "We first generate all multiples of 11 from 100 to 199:", "$$\n110, 121, 132, 143, 154, 165, 176, 187, 198\n$$", "These are the only candidates for $ B $, since $ B $ must be divisible by 11.", "#### Step 2: Identify Valid $ D $ divisible by 9 in [100, 199]", "The smallest and largest multiples of 9 in this range are:", "- $ 108 = 12 \ imes 9 $\n- $ 198 = 22 \ imes 9 $", "Thus, valid $ D $ values are:\n$$\n108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198\n$$", "Note: only multiples of 9 in this interval may trigger a valid $ B $, because $ D $ must satisfy $ 9 \mid D $.", "#### Step 3: For Each $ B \in \ ext{Multiples of 11 in [100, 199]} $, Check Divisibility of Any $ D \in [100,199] $ with $ 9 \mid D $", "We want the smallest such $ B $ such that $ B \mid D $ for some $ D \in [100,199] $, $ 9 \mid D $.", "So, we process candidates in ascending order and test whether they divide any $ D $ in the list above.", "The sorted list of $ B $ candidates:\n$$\n110,\ 121,\ 132,\ 143,\ 154,\ 165,\ 176,\ 187,\ 198\n$$", "We test each $ B $ in order.", "---", "#### Test $ B = 110 $", "Divisors: $ 1, 2, 5, 10, 11, 22, 55, 110 $ → none greater than 198 divisible by 110 except 110 itself, which is less than 100.\nCheck if any $ D \in [100,199] $, divisible by 9, is divisible by 110:\nOnly multiples of 110 in [100,199]: 110 — but 110 not divisible by 9: $ 1+1+0=2 <br/>\not\equiv 0 \pmod{9} $.\nSo no valid $ D $. ✖️", "---", "#### Test $ B = 121 $", "Factors: $ 1, 11, 121 $ — $ 121 = 11^2 $. No multiple of 121 in [100,199] other than 121.\nCheck $ 121 \mid D $ → $ D = 121 $? But $ 1+2+1=4 <br/>\not\equiv 0 \pmod{9} $ → not divisible by 9.\nNo valid $ D $. ✖️", "---", "#### Test $ B = 132 $", "Check divisibility by 11: $ 1 - 3 + 2 = 0 $ → divisible.", "Divisors of 132:\n$ 1, 2, 3, 4, 6, 8, 11, 12, 22, 33, 44, 66, 132 $ → possible $ B $ values in range: all above unless >199 → includes $ 132 $.\nIs $ 132 \mid D $ for some $ D \in [100,199] $, divisible by 9?", "Check multiples of 9 near 132:", "- $ 126 = 14 \ imes 9 $, $ 135 = 15 \ imes 9 $, $ 144 = 16 \ imes 9 $, $ 153 $, $ 162 $, $ 171 $, $ 180 $, $ 189 $, $ 198 $\n- Between 132 and 198 divisible by 9: $ 135, 144, 153, 162, 171, 180, 189, 198 $", "Does 132 divide any of these?", "- $ 132 \mid 132 $? No — 132 is not in list.\n- $ 132 \mid 324 $? Too big.\n- $ 132 \mid 198 $? $ 198 / 132 \approx 1.5 $ → no.", "But wait: we require $ B \mid D $, and $ D \in [100,199] $, $ 9 \mid D $. So we need $ D = B \cdot k $, $ 100 \leq B \cdot k \leq 199 $, $ B \mid D $, $ 9 \mid D $, and $ B $ divisible by 11.", "So reframe: For each $ B $, check whether $ B \cdot k \in [100,199] $, $ 9 \mid (B \cdot k) $", "Start with smallest $ B = 110 $:\n- $ 110 \ imes 1 = 110 $: $ 1+1+0=2 $ → not div by 9\n- $ 110 \ imes 2 = 220 > 199 $ → too big\n→ no valid $ D $", "$ B = 121 $:\n- $ 121 \ imes 1 = 121 $: $ 1+2+1=4 $ → not div by 9\n- $ 121 \ imes 2 = 242 > 199 $ → too big\n→ no", "$ B = 132 $:\n- $ 132 \ imes 1 = 132 $: $ 1+3+2=6 $ → not div by 9\n- $ 132 \ imes 2 = 264 > 199 $ → no", "$ B = 143 $:\n- $ 143 $: $ 1+4+3=8 $ → not div by 9\n- $ 143 \ imes 2 = 286 > 199 $ → no", "$ B = 154 $:\n- $ 154 $: $ 1+5+4=10 $ → no\n- $ 154 \ imes 1 = 154 $: 1+5+4=10 → not div by 9\n- $ 154 \ imes 2 = 308 > 199 $ → no", "$ B = 165 $:\n- $ 165 $: $ 1+6+5=12 $ → not div by 9\n- $ 165 \ imes 1 = 165 $: 12 → no\n- $ 165 \ imes 2 = 330 > 199 $ → no", "$ B = 176 $:\n- $ 176 $: $ 1+7+6=14 $ → no\n- $ 176 \ imes 1 = 176 $: 14 → not div by 9\n- too big otherwise", "$ B = 187 $:\n- $ 187 $: $ 1+8+7=16 $ → no\n- $ 187 \ imes 1 = 187 $: 16 → not div by 9", "$ B = 198 $:\n- $ 198 $: $ 1+9+8=18 $ → divisible by 9! ✅\n- $ 198 \mid D $ → $ D = 198 $?\n- Is $ 198 \in [100,199] $? Yes.\n- $ 9 \mid 198 $? Yes.\n- Is $ 198 $ divisible by 11? $ 1 - 9 + 8 = 0 $ → divisible ✅", "Thus, $ B = 198 $ divides $ D = 198 $, which is in range, divisible by 9, and divisible by 11.", "Even though it’s large, it’s our only candidate that both satisfies $ B \mid D $ and $ 9 \mid D $ and $ B $ divisible by 11. And since we scanned from smallest to largest, 198 is the first such number that satisfies all criteria.", "Wait — could a smaller $ B $ have worked?", "We checked all smaller multiples of 11 — none satisfied the condition. So 198 is the smallest such value.", "But wait — reconsider $ B = 132 $:\nWhat about $ D = 198 $? $ 198 / 132 = 1.5 $ → not integer → no.", "Wait: is there a $ B < 198 $ divisible by 11, in [100,199], dividing some $ D \in [100,199] $, $ 9 \mid D $?", "Let’s test $ B = 121 $ again — $ 121 \mid D $? Only $ D=121 $, sum digits 4 → fails.", "But consider: maybe $ D = 297 $? Too big.", "Wait — what about $ D = 108 $? Is 108 divisible by 11? $ 108 / 11 = 9.81 $ → no.", "Similarly, $ 117 $: $ 1+1+7=9 $ → div by 9. Is $ 117 \mid D $? Only possible $ D = 117 $: $ 1+1+7=9 $ → valid → $ 117 \mid 117 $? Yes. But is 117 divisible by 11? $ 117 \div 11 = 10.63 $ → no.", "Similarly, $ D = 126 $: $ 126 / 11 \approx 11.45 $ → not divisible.", "$ D = 135 $: $ 135 / 11 \approx 12.27 $ → no\n$ D = 144 $: not div by 11\n$ D = 153 $: $ 153 / 11 \approx 13.9 $ → no\n$ D = 162 $: $ 162 / 11 = 14.727 $ → no\n$ D = 171 $: $ 171 / 11 \approx 15.54 $ → no\n$ D = 180 $: $ 180 / 11 \approx 16.36 $ → no\n$ D = 189 $: $ 189 / 11 \approx 17.18 $ → no\n$ D = 198 $: already tested → divisible by"]