But every multiple of 11 could divide such a $ D $, but we want the smallest $ B \equiv 0 \pmod{11} $ such that $ B \mid D $ for some $ D \in [100,199] $, $ 9 \mid D $

["Title: The Smallest Multiple of 11 Dividing a 3-Digit Number in [100,199] with 9-Divisibility", "Meta Description:\nDiscover the smallest number ( B ) such that ( B \equiv 0 \pmod{11} ) and ( B \mid D ) for some three-digit ( D \in [100,199] ) that is divisible by 9. Learn how number theory guides us to this precise divisor.", "---", "### Introduction", "If you’re exploring divisibility, modular arithmetic, and number patterns, you may encounter a compelling question: What is the smallest ( B ) divisible by 11 that divides some number ( D ) in the range [100,199] that is also divisible by 9? This article unravels the logic behind the answer and explains the mathematical reasoning that leads to the unique solution.", "---", "### Understanding the Constraints", "We are looking for:\n- ( B \equiv 0 \pmod{11} ) → ( B ) is a multiple of 11\n- ( B \mid D ) for some ( D \in [100,199] )\n- ( 9 \mid D ) → ( D ) is divisible by 9\n- ( B ) should be minimal satisfying all conditions", "Note: ( B ) does not need to divide all such ( D ), only some ( D ) in [100,199] meeting the divisibility and 9-criteria.", "---", "### Step 1: List the Multiples of 11 in Range [100,199]", "The smallest multiple of 11 in [100,199] is ( 99 \ imes 2 = 198 ), and the largest is ( 11 \ imes 18 = 198 ), then ( 11 \ imes 9 = 99 ) (too low).\nSo valid multiples of 11 in [100,199] are:\n[\n{110, 121, 132, 143, 154, 165, 176, 187, 198}\n]", "But we’re not seeking a multiple of 11, but the smallest such multiple that divides some ( D ) divisible by 9 in the same interval.", "---", "### Step 2: Consider Numbers Divisible by 9 in [100,199]", "A number divisible by 9 has digit sum divisible by 9. List them:", "[\n108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198\n]", "Note: 198 appears in both lists — it is divisible by 11 and 9. But we are not fixing ( D ), we fix ( B ) as smallest multiple of 11 that divides at least one such ( D ).", "---", "### Step 3: Use Divisibility Rules and Number Properties", "Since ( B ) must divide ( D ), and ( D ) is divisible by 9, ( B ) must be compatible with such divisibility. However, ( B ) itself does not need to be divisible by 3 or 11 unless forced — but here it is required to be divisible by 11.", "We are effectively searching for the smallest ( B = 11k ) such that:\n[\nB \leq \max(D) = 199 \quad \ ext{(trivial), and} \quad \exists D \in [100,199],\ 9 \mid D,\ B \mid D\n]", "Ideally, we’d like ( B ) to divide one of the numbers divisible by both 9 and 11 — i.e., divisible by ( \ ext{lcm}(9,11) = 99 ). But ( 99 <br/>\notin [100,199] ), the next multiple is 198 — which is in the list.", "But 198 is divisible by 11 (since ( 198 \div 11 = 18 )), and also divisible by 9 (digit sum = 18), and lies in [100,199]. So does 198 divide itself? Yes.", "Thus, 198 is a candidate, but is it the smallest such multiple of 11 that divides some multiple of 9 in [100,199]?", "Check smaller multiples of 11 from the list:", "- ( 110 ): Does 110 divide any multiple of 9 in [100,199]?\n Next multiple of 9: 108 → ( 108 \div 110 < 1 ), not divisible.\n Next: 117, 126… 120 → not divisible.\n Check: ( 11 \ imes 10 = 110 ). Is there ( D = 110k \in [100,199] )?\n ( 110 \ imes 1 = 110 <br/>\not\in [100,199] ), and next is ( 110 \ imes 2 = 220 > 199 ). So no such ( D ).\n So 110 does not divide any ( D ) in [100,199] divisible by 9.", "- ( 121 ): next multiple: 121×1=121, does 121×2=242 > 199 → only candidate is 121.\n Check if 121 divides any multiple of 9 in range:\n ( 121 \ imes 1 = 121 ): not divisible by 9 (1+2+1=4)\n ( 121 \ imes 2 = 242 > 199 ) → no.", "- ( 132 ): check if any multiple of 9 in [100,199] divisible by 132.\n ( 132 \ imes 1 = 132 ), check if 132 divisible by 9? 1+3+2=6 → no.\n Next multiple: ( 132 \ imes 2 = 264 > 199 ). So no.", "- ( 143 ): divisible by 11? 1+4+3=8 → no. But does it divide a multiple of 9 in range?\n Multiples of 9: 135 → ( 135 \div 143 < 1 ), 144: 144 < 143? No. So no multiple of 9 divisible by 143 in range.", "- ( 154 ): ( 154 \div 2 = 77 ), is it divisible by 9? 1+5+4=10 → no.\n 154×1=154 (not div by 9), 308 > 199 → no.", "- ( 165 ): divisible by 11? 1+6+5=12 → no.\n ( 165 \div 9 = 18.3 ), not integer → no.", "- ( 176 ): ( 1+7+6=14 ) → not divisible by 9. Not multiple of 9.", "- ( 187 ): ( 1+8+7=16 ) → not divisible by 9.", "- ( 198 ): divisible by 11? ( 198 \div 11 = 18 ) ✅\n Divisible by 9? ( 1+9+8=18 ) ✅\n And: ( 198 \mid 198 ), and 198 ∈ [100,199] ✅", "So far, 198 is the only multiple of 11 in [100,199] that divides a multiple of 9 in the same interval.", "But wait: could a smaller multiple of 11 divide a multiple of 9 (not equal to itself) in [100,199]?", "Try ( B = 132 ): We checked — does any multiple of 9 divisible by 132?\nMultiples near 132: 108, 162, 198\n162 ÷ 132 ≈ 1.227 → not divisible\n198 ÷ 132 ≈ 1.5 → not integer → no.", "Try ( B = 121 ):\nMultiples of 9: 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198\nCheck divisibility:\n171 ÷ 121 ≈ 1.41 → no\n189 ÷ 121 ≈ 1.56 → no\n198 ÷ 121 ≈ 1.636 → no\nNone divisible by 121.", "Try ( B = 110 ):\n108, 117, ..., 180:\n180 ÷ 110 ≈ 1.63 → no\n108 ÷ 110 < 1 → no\n150? Not divisible by 9.\nNo.", "Try ( B = 99 ): Not divisible by 11 → excluded.", "Try ( B = 22 ): too small, not divisible by 11.", "Wait — are we missing something?", "Suppose ( B = 132 ), and suppose ( D = 396 ) — but ( D = 396 > 199 ), outside range.", "So range is strict: [100,199], so only multiples of 9 and 11 in this window.", "But here’s the key: Is there any ( B < 198 ) divisible by 11, such that ( B \mid D ) for some D divisible by 9 in [100,199]?", "Suppose such a ( B = 11k ) with ( B < 198 ), ( D = 11k \mid D ), ( B \mid D ), ( D \in [100,199] ), divisible by 9.", "Then ( D ) must be a multiple of ( B ) and divisible by 9. So ( D = \ ext{lcm}(B, 9) \cdot m ), but within [100,199].", "But if ( B ) is small, say 99, but 99 not divisible by 11 → invalid.", "Next: ( B = 33, 66, 99, 132, 165, 198 ) — none less than 198 and divisible by 11 except 132 and 165.", "We checked 132 and 165: none divide any multiple of 9 in [100,199].", "Therefore, 198 is the only candidate in the range that is divisible by 11 and divides a multiple (itself) of 9 in [100,199].", "But wait — could a smaller multiple of 11 divide a multiple of 9?", "For example, could ( B = 11 \ imes 1 = 11 )? Too small — not in [100,199].", "But what if ( D ) is a multiple of both 11 and 9 — i.e., divisible by 99 — only 198 qualifies in [100,199].", "So any ( B ) dividing such a ( D ) must divide 198 (the least common multiple); and divisors of 198 in [100,199] include only 198.", "Therefore, 198 is the unique such number"]