{
  "index": "1956-A-2",
  "type": "NT",
  "tag": [
    "NT",
    "COMB"
  ],
  "difficulty": "",
  "question": "2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.",
  "solution": "Solution. If \\( n \\) is a positive integer and \\( p \\) is any other positive integer, then one of the integers\n\\[\np+1, p+2, \\ldots, p+n\n\\]\nis a multiple of \\( n \\). Given \\( n \\), choose \\( p=1,234,567,890 \\times 10^{k} \\), where \\( k \\) is so large that \\( 10^{k}>n \\). Then all of the integers \\( p+1, p+2, \\ldots, p+n \\) have decimal representations beginning with \\( 1234567890 \\ldots \\), and one of these is a multiple of \\( n \\).",
  "vars": [
    "n",
    "p",
    "k"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "targetnum",
        "p": "candidate",
        "k": "exponent"
      },
      "question": "2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.",
      "solution": "Solution. If \\( targetnum \\) is a positive integer and \\( candidate \\) is any other positive integer, then one of the integers\n\\[\ncandidate+1, candidate+2, \\ldots, candidate+targetnum\n\\]\nis a multiple of \\( targetnum \\). Given \\( targetnum \\), choose \\( candidate=1,234,567,890 \\times 10^{exponent} \\), where \\( exponent \\) is so large that \\( 10^{exponent}>targetnum \\). Then all of the integers \\( candidate+1, candidate+2, \\ldots, candidate+targetnum \\) have decimal representations beginning with \\( 1234567890 \\ldots \\), and one of these is a multiple of \\( targetnum \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "tangerine",
        "p": "cloudburst",
        "k": "lumberjack"
      },
      "question": "2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.",
      "solution": "Solution. If \\( tangerine \\) is a positive integer and \\( cloudburst \\) is any other positive integer, then one of the integers\n\\[\ncloudburst+1, cloudburst+2, \\ldots, cloudburst+tangerine\n\\]\nis a multiple of \\( tangerine \\). Given \\( tangerine \\), choose \\( cloudburst=1,234,567,890 \\times 10^{lumberjack} \\), where \\( lumberjack \\) is so large that \\( 10^{lumberjack}>tangerine \\). Then all of the integers \\( cloudburst+1, cloudburst+2, \\ldots, cloudburst+tangerine \\) have decimal representations beginning with \\( 1234567890 \\ldots \\), and one of these is a multiple of \\( tangerine \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "negativenumber",
        "p": "specificnegative",
        "k": "smallindex"
      },
      "question": "2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.",
      "solution": "Solution. If \\( negativenumber \\) is a positive integer and \\( specificnegative \\) is any other positive integer, then one of the integers\n\\[\nspecificnegative+1, specificnegative+2, \\ldots, specificnegative+negativenumber\n\\]\nis a multiple of \\( negativenumber \\). Given \\( negativenumber \\), choose \\( specificnegative=1,234,567,890 \\times 10^{smallindex} \\), where \\( smallindex \\) is so large that \\( 10^{smallindex}>negativenumber \\). Then all of the integers \\( specificnegative+1, specificnegative+2, \\ldots, specificnegative+negativenumber \\) have decimal representations beginning with \\( 1234567890 \\ldots \\), and one of these is a multiple of \\( negativenumber \\)."
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "p": "hjgrksla",
        "k": "vcmqptne"
      },
      "question": "2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.",
      "solution": "Solution. If \\( qzxwvtnp \\) is a positive integer and \\( hjgrksla \\) is any other positive integer, then one of the integers\n\\[\nhjgrksla+1, hjgrksla+2, \\ldots, hjgrksla+qzxwvtnp\n\\]\nis a multiple of \\( qzxwvtnp \\). Given \\( qzxwvtnp \\), choose \\( hjgrksla=1,234,567,890 \\times 10^{vcmqptne} \\), where \\( vcmqptne \\) is so large that \\( 10^{vcmqptne}>qzxwvtnp \\). Then all of the integers \\( hjgrksla+1, hjgrksla+2, \\ldots, hjgrksla+qzxwvtnp \\) have decimal representations beginning with \\( 1234567890 \\ldots \\), and one of these is a multiple of \\( qzxwvtnp \\)."
    },
    "kernel_variant": {
      "question": "Let $n$ be a positive integer such that  \n\\[\n\\gcd(n,30)=1 ,\n\\]  \nand fix an arbitrary residue vector  \n\\[\n\\bigl(r_{0},r_{1},\\dots ,r_{9}\\bigr)\\in\\bigl(\\mathbf Z/n\\mathbf Z\\bigr)^{10}.\n\\]  \nProve that there exists a positive integer $N$ with the following three properties.\n\n1. $N\\equiv 0 \\pmod n$;\n\n2. the ordinary (base–$10$) decimal expansion of $N$ begins with the ten–digit block  \n   \\[\n     \\boxed{\\,907\\,856\\,3412\\,};\n   \\]\n\n3. for every decimal digit $d$ $(0\\le d\\le 9)$ the total number of\n   occurrences of $d$ in the full decimal expansion of $N$ is congruent to $r_{d}\\pmod n$.\n\n(In particular, choosing $r_{0}=r_{1}=\\dots =r_{9}=1$ delivers a pandigital multiple of $n$ that starts with the required prefix.)\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "All congruences are taken modulo $n$ unless explicitly stated otherwise.\n\n--------------------------------------------------------------------\nA.  The $10$–adic period\n--------------------------------------------------------------------\nSince $\\gcd(n,10)=1$, the multiplicative order  \n\\[\nm:=\\operatorname{ord}_{\\,n}(10)\\qquad\\bigl(10^{m}\\equiv1\\bigr)\n\\]\nis well–defined.  Only the consequence $10^{m}\\equiv1$ will be needed.\n\n--------------------------------------------------------------------\nB.  $m$–blocks and bookkeeping parameters\n--------------------------------------------------------------------\nFor $0\\le d\\le9$ and $0\\le j\\le m-1$ let $B_{d,j}$ be the word of\nlength $m$ whose only (possibly) non–zero digit is $d$, situated $j$\nplaces from the right:\n\\[\nB_{d,j}=\\underbrace{0\\dots0}_{m-j-1}\\,d\\,\\underbrace{0\\dots0}_{j}.\n\\]\nIntroduce non–negative integers  \n\\[\nx_{d,j}\\in\\mathbf Z_{\\ge0}\\qquad(d=0,\\dots ,9,\\;j=0,\\dots ,m-1)\n\\]\nwhich specify how many copies of $B_{d,j}$ will be used, and set  \n\n\\[\nC_{d}:=\\sum_{j=0}^{m-1}x_{d,j},\\qquad    \nL:=\\sum_{d=0}^{9}C_{d}=\\sum_{d,j}x_{d,j}.\n\\]\n\nThe word $Y$ obtained by concatenation of all the chosen blocks therefore satisfies  \n\n\\[\n|Y|=mL,\\qquad\n\\#d(Y)=C_{d}\\quad(1\\le d\\le9),\n\\]\nwhile the digit $0$ occurs in every block that carries a non–zero digit plus in the $C_{0}$ own blocks, giving  \n\\[\n\\#0(Y)=(m-1)L+C_{0}.                                     \\tag{1}\n\\]\n\nFinally the numerical value of $Y$ is  \n\\[\nV:=\\operatorname{value}(Y)\\equiv\n      \\sum_{d=1}^{9}\\sum_{j=0}^{m-1}x_{d,j}\\,d\\,10^{j}.   \\tag{2}\n\\]\n\n--------------------------------------------------------------------\nC.  Choosing the digit frequencies\n--------------------------------------------------------------------\nLet the prescribed ten–digit prefix be  \n\\[\nP:=907\\,856\\,3412 ,\\qquad U:=\\operatorname{value}(P);\n\\]\nits digit–count vector is $(1,1,\\dots ,1)$.  Define  \n\\[\n\\kappa_{d}:=r_{d}-1\\quad(d=0,\\dots ,9),\\qquad\n\\xi\\equiv-U\\pmod n.                                      \\tag{3}\n\\]\n\nStep C1.  Fix arbitrarily  \n\\[\n\\widehat C_{d}\\equiv\\kappa_{d}\\pmod n\\qquad(1\\le d\\le9)\n\\]\nand impose the \\emph{size condition}  \n\\[\n\\widehat C_{d}\\ge n\\,m\\qquad(1\\le d\\le9).                \\tag{4}\n\\]\n(The freedom to choose the $\\widehat C_{d}$ arbitrarily high makes this possible.)\n\nStep C2.  Put  \n\\[\nS:=\\sum_{d=1}^{9}\\widehat C_{d}\\equiv\n       \\sum_{d=1}^{9}\\kappa_{d}\\pmod n .\n\\]\nSelect an integer $\\widehat L$ with  \n\\[\n\\widehat L\\equiv S\\pmod n,\\qquad \\widehat L\\ge n\\,m,      \\tag{5}\n\\]\nand determine  \n\\[\n\\widehat C_{0}\\equiv\\kappa_{0}-(m-1)\\widehat L\\pmod n,\n\\qquad \\widehat C_{0}\\ge n\\,m.                            \\tag{6}\n\\]\nBecause of (1) we obtain  \n\\[\n\\#d(Y)\\equiv\\kappa_{d}\\pmod n\\qquad(0\\le d\\le9).          \\tag{7}\n\\]\n\n--------------------------------------------------------------------\nD.  A solvable linear condition for the value $V$\n--------------------------------------------------------------------\nPlace \\emph{all} $\\widehat C_{d}$ copies of digit $d$\ntemporarily at position $j=0$.  With  \n\\[\nV_{0}:=\\sum_{d=1}^{9}\\widehat C_{d}\\,d ,                     \\tag{8}\n\\]\nwe have $V\\equiv V_{0}$.  To achieve the required residue  \n\\[\nV\\equiv\\xi\\pmod n ,                                        \\tag{9}\n\\]\nwe only move the digit $1$.\n\nLet  \n\\[\n\\Delta:=\\xi-V_{0}\\pmod n .\n\\]\n\nLemma (Generation).  Because $\\gcd(n,9)=1$, the set  \n\\[\n\\bigl\\{\\,10^{j}-1\\mid1\\le j\\le m-1\\bigr\\}\\subset\\mathbf Z/n\\mathbf Z\n\\]\ngenerates the whole additive group $\\mathbf Z/n\\mathbf Z$.\n\nProof.  If a divisor $g$ of $n$ annihilated every $10^{j}-1$, then\n$10^{j}\\equiv1\\pmod g$ for all $j$ and hence $10\\equiv1\\pmod g$,\nso $9\\equiv0\\pmod g$.  As $\\gcd(n,9)=1$, this forces $g=1$.\n\nConsequently there exist non–negative integers $t_{1},\\dots ,t_{m-1}$\nsuch that  \n\\[\n\\sum_{j=1}^{m-1}t_{j}(10^{j}-1)\\equiv\\Delta\\pmod n        \\tag{10}\n\\]\nand  \n\\[\nT:=\\sum_{j=1}^{m-1}t_{j}\\le n(m-1).                       \\tag{11}\n\\]\n\nRedistribution step.\n\n• Remove $T$ copies of digit $1$ from position $j=0$;\n\n• insert $t_{j}$ copies of digit $1$ at every position $j=1,\\dots ,m-1$.\n\nSince $T=\\sum t_{j}$, the total count of digit $1$ remains $\\widehat C_{1}$ and\n(10) guarantees (9).  All other digits stay untouched, hence (7) still holds.\n\nNon–negativity of all $x_{d,j}$.  \nOnly the multiplicity $x_{1,0}$ is potentially diminished:\n\\[\nx_{1,0}=\\widehat C_{1}-T\n        \\stackrel{(11)}{\\ge}\\widehat C_{1}-n(m-1)\n        \\stackrel{(4)}{\\ge}n\\,m-n(m-1)=n\\ge0 .\n\\]\nTherefore every $x_{d,j}$ is non–negative, and in particular $x_{1,0}\\ge n>0$.\n\n--------------------------------------------------------------------\nE.  Building the final number\n--------------------------------------------------------------------\nLet $Y$ be the $m$–block word just obtained and set  \n\\[\n\\ell:=|Y|=m\\widehat L,\\qquad\nN:=U\\cdot10^{\\ell}+V.\n\\]\n\n(i)  Because $10^{\\ell}\\equiv1$, relations (3) and (9) yield  \n\\[\nN\\equiv U+\\xi\\equiv0\\pmod n .\n\\]\n\n(ii)  Since $V<10^{\\ell}$, no carrying occurs when $V$ is added to\n$U\\cdot10^{\\ell}$; hence the decimal expansion of $N$ does indeed start with $P$.\n\n(iii)  Adding the digit counts in $P$ to those in $Y$ and using (7) gives  \n\\[\n\\#d(N)=\\#d(P)+\\#d(Y)\\equiv1+\\kappa_{d}\\equiv r_{d}\\pmod n\n       \\qquad(0\\le d\\le9).\n\\]\n\nThus $N$ fulfils all three required properties. \\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.476850",
        "was_fixed": false,
        "difficulty_analysis": "1.  The original problem only asked for a pandigital multiple; the present variant prescribes an independent congruence\n    (#d) ≡ r_d (mod n) for each of the ten digits, so ten simultaneous modular conditions must be met in addition to the divisibility by n and the fixed prefix.  \n\n2.  Meeting these conditions forces the solver to juggle three distinct structures at once:\n    • combinatorial control of digit–frequencies (Step 1),  \n    • preservation of those frequencies while still retaining freedom to vary remainders (Step 2), and  \n    • a non-trivial recurrence and a case-by-case modular argument (Step 3) to ensure one of the constructed numbers is actually a multiple of n.  \n\n3.  The neutral block W and the recurrence (7) are new technical devices that do not appear in the basic kernel problem; analysing them requires a blend of combinatorial counting, modular arithmetic, and discrete dynamical systems.  \n\n4.  The proof must contend with arbitrary common factors between the integer base 10 and n, something that the original argument avoided by simply taking a sufficiently large power of 10.  Handling the general case results in the dichotomy “n divides b” vs. “n and b are coprime”, and forces a careful greatest-common-divisor analysis.  \n\n5.  In short, the enhanced variant replaces a single pigeon-hole step by a multi-layer construction that simultaneously solves ten independent congruence conditions and a divisibility constraint, pushing the solver into significantly deeper modular-combinatorial territory."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n$ be a positive integer such that  \n\\[\n\\gcd(n,30)=1 ,\n\\]  \nand fix an arbitrary residue vector  \n\\[\n\\bigl(r_{0},r_{1},\\dots ,r_{9}\\bigr)\\in\\bigl(\\mathbf Z/n\\mathbf Z\\bigr)^{10}.\n\\]  \nProve that there exists a positive integer $N$ with the following three properties.\n\n1. $N\\equiv 0 \\pmod n$;\n\n2. the ordinary (base–$10$) decimal expansion of $N$ begins with the ten–digit block  \n   \\[\n     \\boxed{\\,907\\,856\\,3412\\,};\n   \\]\n\n3. for every decimal digit $d$ $(0\\le d\\le 9)$ the total number of\n   occurrences of $d$ in the full decimal expansion of $N$ is congruent to $r_{d}\\pmod n$.\n\n(In particular, choosing $r_{0}=r_{1}=\\dots =r_{9}=1$ delivers a pandigital multiple of $n$ that starts with the required prefix.)\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "All congruences are taken modulo $n$ unless explicitly stated otherwise.\n\n--------------------------------------------------------------------\nA.  The $10$–adic period\n--------------------------------------------------------------------\nSince $\\gcd(n,10)=1$, the multiplicative order  \n\\[\nm:=\\operatorname{ord}_{\\,n}(10)\\qquad\\bigl(10^{m}\\equiv1\\bigr)\n\\]\nis well–defined.  Only the consequence $10^{m}\\equiv1$ will be needed.\n\n--------------------------------------------------------------------\nB.  $m$–blocks and bookkeeping parameters\n--------------------------------------------------------------------\nFor $0\\le d\\le9$ and $0\\le j\\le m-1$ let $B_{d,j}$ be the word of\nlength $m$ whose only (possibly) non–zero digit is $d$, situated $j$\nplaces from the right:\n\\[\nB_{d,j}=\\underbrace{0\\dots0}_{m-j-1}\\,d\\,\\underbrace{0\\dots0}_{j}.\n\\]\nIntroduce non–negative integers  \n\\[\nx_{d,j}\\in\\mathbf Z_{\\ge0}\\qquad(d=0,\\dots ,9,\\;j=0,\\dots ,m-1)\n\\]\nwhich specify how many copies of $B_{d,j}$ will be used, and set  \n\n\\[\nC_{d}:=\\sum_{j=0}^{m-1}x_{d,j},\\qquad    \nL:=\\sum_{d=0}^{9}C_{d}=\\sum_{d,j}x_{d,j}.\n\\]\n\nThe word $Y$ obtained by concatenation of all the chosen blocks therefore satisfies  \n\n\\[\n|Y|=mL,\\qquad\n\\#d(Y)=C_{d}\\quad(1\\le d\\le9),\n\\]\nwhile the digit $0$ occurs in every block that carries a non–zero digit plus in the $C_{0}$ own blocks, giving  \n\\[\n\\#0(Y)=(m-1)L+C_{0}.                                     \\tag{1}\n\\]\n\nFinally the numerical value of $Y$ is  \n\\[\nV:=\\operatorname{value}(Y)\\equiv\n      \\sum_{d=1}^{9}\\sum_{j=0}^{m-1}x_{d,j}\\,d\\,10^{j}.   \\tag{2}\n\\]\n\n--------------------------------------------------------------------\nC.  Choosing the digit frequencies\n--------------------------------------------------------------------\nLet the prescribed ten–digit prefix be  \n\\[\nP:=907\\,856\\,3412 ,\\qquad U:=\\operatorname{value}(P);\n\\]\nits digit–count vector is $(1,1,\\dots ,1)$.  Define  \n\\[\n\\kappa_{d}:=r_{d}-1\\quad(d=0,\\dots ,9),\\qquad\n\\xi\\equiv-U\\pmod n.                                      \\tag{3}\n\\]\n\nStep C1.  Fix arbitrarily  \n\\[\n\\widehat C_{d}\\equiv\\kappa_{d}\\pmod n\\qquad(1\\le d\\le9)\n\\]\nand impose the \\emph{size condition}  \n\\[\n\\widehat C_{d}\\ge n\\,m\\qquad(1\\le d\\le9).                \\tag{4}\n\\]\n(The freedom to choose the $\\widehat C_{d}$ arbitrarily high makes this possible.)\n\nStep C2.  Put  \n\\[\nS:=\\sum_{d=1}^{9}\\widehat C_{d}\\equiv\n       \\sum_{d=1}^{9}\\kappa_{d}\\pmod n .\n\\]\nSelect an integer $\\widehat L$ with  \n\\[\n\\widehat L\\equiv S\\pmod n,\\qquad \\widehat L\\ge n\\,m,      \\tag{5}\n\\]\nand determine  \n\\[\n\\widehat C_{0}\\equiv\\kappa_{0}-(m-1)\\widehat L\\pmod n,\n\\qquad \\widehat C_{0}\\ge n\\,m.                            \\tag{6}\n\\]\nBecause of (1) we obtain  \n\\[\n\\#d(Y)\\equiv\\kappa_{d}\\pmod n\\qquad(0\\le d\\le9).          \\tag{7}\n\\]\n\n--------------------------------------------------------------------\nD.  A solvable linear condition for the value $V$\n--------------------------------------------------------------------\nPlace \\emph{all} $\\widehat C_{d}$ copies of digit $d$\ntemporarily at position $j=0$.  With  \n\\[\nV_{0}:=\\sum_{d=1}^{9}\\widehat C_{d}\\,d ,                     \\tag{8}\n\\]\nwe have $V\\equiv V_{0}$.  To achieve the required residue  \n\\[\nV\\equiv\\xi\\pmod n ,                                        \\tag{9}\n\\]\nwe only move the digit $1$.\n\nLet  \n\\[\n\\Delta:=\\xi-V_{0}\\pmod n .\n\\]\n\nLemma (Generation).  Because $\\gcd(n,9)=1$, the set  \n\\[\n\\bigl\\{\\,10^{j}-1\\mid1\\le j\\le m-1\\bigr\\}\\subset\\mathbf Z/n\\mathbf Z\n\\]\ngenerates the whole additive group $\\mathbf Z/n\\mathbf Z$.\n\nProof.  If a divisor $g$ of $n$ annihilated every $10^{j}-1$, then\n$10^{j}\\equiv1\\pmod g$ for all $j$ and hence $10\\equiv1\\pmod g$,\nso $9\\equiv0\\pmod g$.  As $\\gcd(n,9)=1$, this forces $g=1$.\n\nConsequently there exist non–negative integers $t_{1},\\dots ,t_{m-1}$\nsuch that  \n\\[\n\\sum_{j=1}^{m-1}t_{j}(10^{j}-1)\\equiv\\Delta\\pmod n        \\tag{10}\n\\]\nand  \n\\[\nT:=\\sum_{j=1}^{m-1}t_{j}\\le n(m-1).                       \\tag{11}\n\\]\n\nRedistribution step.\n\n• Remove $T$ copies of digit $1$ from position $j=0$;\n\n• insert $t_{j}$ copies of digit $1$ at every position $j=1,\\dots ,m-1$.\n\nSince $T=\\sum t_{j}$, the total count of digit $1$ remains $\\widehat C_{1}$ and\n(10) guarantees (9).  All other digits stay untouched, hence (7) still holds.\n\nNon–negativity of all $x_{d,j}$.  \nOnly the multiplicity $x_{1,0}$ is potentially diminished:\n\\[\nx_{1,0}=\\widehat C_{1}-T\n        \\stackrel{(11)}{\\ge}\\widehat C_{1}-n(m-1)\n        \\stackrel{(4)}{\\ge}n\\,m-n(m-1)=n\\ge0 .\n\\]\nTherefore every $x_{d,j}$ is non–negative, and in particular $x_{1,0}\\ge n>0$.\n\n--------------------------------------------------------------------\nE.  Building the final number\n--------------------------------------------------------------------\nLet $Y$ be the $m$–block word just obtained and set  \n\\[\n\\ell:=|Y|=m\\widehat L,\\qquad\nN:=U\\cdot10^{\\ell}+V.\n\\]\n\n(i)  Because $10^{\\ell}\\equiv1$, relations (3) and (9) yield  \n\\[\nN\\equiv U+\\xi\\equiv0\\pmod n .\n\\]\n\n(ii)  Since $V<10^{\\ell}$, no carrying occurs when $V$ is added to\n$U\\cdot10^{\\ell}$; hence the decimal expansion of $N$ does indeed start with $P$.\n\n(iii)  Adding the digit counts in $P$ to those in $Y$ and using (7) gives  \n\\[\n\\#d(N)=\\#d(P)+\\#d(Y)\\equiv1+\\kappa_{d}\\equiv r_{d}\\pmod n\n       \\qquad(0\\le d\\le9).\n\\]\n\nThus $N$ fulfils all three required properties. \\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.400826",
        "was_fixed": false,
        "difficulty_analysis": "1.  The original problem only asked for a pandigital multiple; the present variant prescribes an independent congruence\n    (#d) ≡ r_d (mod n) for each of the ten digits, so ten simultaneous modular conditions must be met in addition to the divisibility by n and the fixed prefix.  \n\n2.  Meeting these conditions forces the solver to juggle three distinct structures at once:\n    • combinatorial control of digit–frequencies (Step 1),  \n    • preservation of those frequencies while still retaining freedom to vary remainders (Step 2), and  \n    • a non-trivial recurrence and a case-by-case modular argument (Step 3) to ensure one of the constructed numbers is actually a multiple of n.  \n\n3.  The neutral block W and the recurrence (7) are new technical devices that do not appear in the basic kernel problem; analysing them requires a blend of combinatorial counting, modular arithmetic, and discrete dynamical systems.  \n\n4.  The proof must contend with arbitrary common factors between the integer base 10 and n, something that the original argument avoided by simply taking a sufficiently large power of 10.  Handling the general case results in the dichotomy “n divides b” vs. “n and b are coprime”, and forces a careful greatest-common-divisor analysis.  \n\n5.  In short, the enhanced variant replaces a single pigeon-hole step by a multi-layer construction that simultaneously solves ten independent congruence conditions and a divisibility constraint, pushing the solver into significantly deeper modular-combinatorial territory."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}