{
  "index": "1975-B-1",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "B-1. In the additive group of ordered pairs of integers ( \\( m, n \\) ) [with addition defined componentwise: \\( \\left.(m, n)+\\left(m^{\\prime}, n^{\\prime}\\right)=\\left(m+m^{\\prime}, n+n^{\\prime}\\right)\\right] \\) consider the subgroup \\( H \\) generated by the three elements\n\\[\n(3,8), \\quad(4,-1)\n\\]\n\nThen \\( \\boldsymbol{H} \\) has another set of generators of the form\n\\[\n(1, b), \\quad(0, a)\n\\]\nfor some integers \\( a, b \\) with \\( a>0 \\). Find \\( a \\).\n[Elements \\( g_{1}, \\ldots, g_{k} \\) are said to generate a subgroup \\( H \\) if (i) each \\( g_{1} \\in H \\), and (ii) every \\( h \\in H \\) can be written as a sum \\( h=n_{1} g_{1}+\\cdots+n_{k} g_{k} \\) where the \\( n_{1} \\) are integers (and where, for example, \\( 3 g_{1}-2 g_{2} \\) means \\( \\left.\\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\\right).\\right] \\)",
  "solution": "B-1.\nThe answer is \\( a=7 \\). Also one must have \\( b \\equiv 5(\\bmod 7) \\).\nProof: The subgroup \\( H \\) must contain \\( 4(3,8)-3(4,-1)=(0,35), 4(5,4)-5(4,-1)=(0,21) \\), and then \\( 2(0,21)-(0,35)=(0,7) \\). Now \\( (0,7) \\) and \\( (1, b) \\) will generate \\( H \\) iff \\( (1, b) \\) is in \\( H \\) and there exist integers \\( u \\), \\( v \\), and \\( w \\) such that\n\\[\n(3,8)=3(1, b)+u(0,7),(4,-1)=4(1, b)+v(0,7),(5,4)=5(1, b)+w(0,7)\n\\]\n\nThese hold iff \\( 8=3 b+7 u,-1=4 b+7 v \\), and \\( 4=5 b+7 w \\). With \\( b=5+7 k, k \\) any integer, the desired coefficients \\( u, v \\), and \\( w \\) exist in the form \\( u=-1-3 k, v=-3-4 k, w=-3-5 k \\). It now suffices to let \\( k=0 \\) and to note that \\( (1,5)=(4,-1)-(3,8)+2(0,7) \\) is in \\( H \\).",
  "vars": [
    "m",
    "n",
    "n_1",
    "g_1",
    "g_2",
    "g_k",
    "u",
    "v",
    "w",
    "k"
  ],
  "params": [
    "a",
    "b",
    "H"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "m": "firstcoord",
        "n": "secondcoord",
        "n_1": "coeffone",
        "g_1": "generatorone",
        "g_2": "generatortwo",
        "g_k": "generatork",
        "u": "integeru",
        "v": "integerv",
        "w": "integerw",
        "k": "parameterk",
        "a": "stepvalue",
        "b": "shiftvalue",
        "H": "subgrouph"
      },
      "question": "B-1. In the additive group of ordered pairs of integers ( \\( firstcoord, secondcoord \\) ) [with addition defined componentwise: \\( \\left.(firstcoord, secondcoord)+\\left(firstcoord^{\\prime}, secondcoord^{\\prime}\\right)=\\left(firstcoord+firstcoord^{\\prime}, secondcoord+secondcoord^{\\prime}\\right)\\right] \\) consider the subgroup \\( subgrouph \\) generated by the three elements\n\\[\n(3,8), \\quad(4,-1)\n\\]\n\nThen \\( \\boldsymbol{subgrouph} \\) has another set of generators of the form\n\\[\n(1, shiftvalue), \\quad(0, stepvalue)\n\\]\nfor some integers \\( stepvalue, shiftvalue \\) with \\( stepvalue>0 \\). Find \\( stepvalue \\).\n[Elements \\( generatorone, \\ldots, generatork \\) are said to generate a subgroup \\( subgrouph \\) if (i) each \\( generatorone \\in subgrouph \\), and (ii) every \\( h \\in subgrouph \\) can be written as a sum \\( h=coeffone\\,generatorone+\\cdots+n_{k}\\,generatork \\) where the \\( coeffone \\) are integers (and where, for example, \\( 3\\,generatorone-2\\,generatortwo \\) means \\( \\left.\\left.generatorone+generatorone+generatorone-generatortwo-generatortwo\\right).\\right] )\"",
      "solution": "B-1.\nThe answer is \\( stepvalue=7 \\). Also one must have \\( shiftvalue \\equiv 5(\\bmod 7) \\).\n\nProof: The subgroup \\( subgrouph \\) must contain \\( 4(3,8)-3(4,-1)=(0,35), 4(5,4)-5(4,-1)=(0,21) \\), and then \\( 2(0,21)-(0,35)=(0,7) \\). Now \\( (0,7) \\) and \\( (1, shiftvalue) \\) will generate \\( subgrouph \\) iff \\( (1, shiftvalue) \\) is in \\( subgrouph \\) and there exist integers \\( integeru \\), \\( integerv \\), and \\( integerw \\) such that\n\\[\n(3,8)=3(1, shiftvalue)+integeru(0,7),\\quad(4,-1)=4(1, shiftvalue)+integerv(0,7),\\quad(5,4)=5(1, shiftvalue)+integerw(0,7)\n\\]\n\nThese hold iff \\( 8=3\\,shiftvalue+7\\,integeru,\\,-1=4\\,shiftvalue+7\\,integerv \\), and \\( 4=5\\,shiftvalue+7\\,integerw \\). With \\( shiftvalue=5+7\\,parameterk,\\; parameterk \\) any integer, the desired coefficients \\( integeru, integerv \\), and \\( integerw \\) exist in the form \\( integeru=-1-3\\,parameterk,\\; integerv=-3-4\\,parameterk,\\; integerw=-3-5\\,parameterk \\). It now suffices to let \\( parameterk=0 \\) and to note that \\( (1,5)=(4,-1)-(3,8)+2(0,7) \\) is in \\( subgrouph \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "m": "pineapple",
        "n": "grapefruit",
        "n_1": "tangerine",
        "g_1": "marigold",
        "g_2": "salamander",
        "g_k": "hummingbird",
        "u": "blackbird",
        "v": "gingerroot",
        "w": "butterscotch",
        "k": "cheesecake",
        "a": "rainstorm",
        "b": "moonlight",
        "H": "caterpillar"
      },
      "question": "B-1. In the additive group of ordered pairs of integers ( \\( pineapple, grapefruit \\) ) [with addition defined componentwise: \\( \\left.(pineapple, grapefruit)+\\left(pineapple^{\\prime}, grapefruit^{\\prime}\\right)=\\left(pineapple+pineapple^{\\prime}, grapefruit+grapefruit^{\\prime}\\right)\\right] \\) consider the subgroup \\( caterpillar \\) generated by the three elements\n\\[\n(3,8), \\quad(4,-1)\n\\]\n\nThen \\( \\boldsymbol{caterpillar} \\) has another set of generators of the form\n\\[\n(1, moonlight), \\quad(0, rainstorm)\n\\]\nfor some integers \\( rainstorm, moonlight \\) with \\( rainstorm>0 \\). Find \\( rainstorm \\).\n[Elements \\( marigold, \\ldots, hummingbird \\) are said to generate a subgroup \\( caterpillar \\) if (i) each \\( marigold \\in caterpillar \\), and (ii) every \\( h \\in caterpillar \\) can be written as a sum \\( h=tangerine\\,marigold+\\cdots+n_{k}\\,hummingbird \\) where the \\( tangerine \\) are integers (and where, for example, \\( 3\\,marigold-2\\,salamander \\) means \\( \\left.\\left.marigold+marigold+marigold-salamander-salamander\\right).\\right] \\)",
      "solution": "B-1.\nThe answer is \\( rainstorm=7 \\). Also one must have \\( moonlight \\equiv 5(\\bmod 7) \\).\n\nProof: The subgroup \\( caterpillar \\) must contain \\( 4(3,8)-3(4,-1)=(0,35), 4(5,4)-5(4,-1)=(0,21) \\), and then \\( 2(0,21)-(0,35)=(0,7) \\). Now \\( (0,7) \\) and \\( (1, moonlight) \\) will generate \\( caterpillar \\) iff \\( (1, moonlight) \\) is in \\( caterpillar \\) and there exist integers \\( blackbird \\), \\( gingerroot \\), and \\( butterscotch \\) such that\n\\[\n(3,8)=3(1, moonlight)+blackbird(0,7),\\quad(4,-1)=4(1, moonlight)+gingerroot(0,7),\\quad(5,4)=5(1, moonlight)+butterscotch(0,7)\n\\]\n\nThese hold iff \\( 8=3\\,moonlight+7\\,blackbird,\\,-1=4\\,moonlight+7\\,gingerroot \\), and \\( 4=5\\,moonlight+7\\,butterscotch \\). With \\( moonlight=5+7\\,cheesecake,\\;cheesecake \\) any integer, the desired coefficients \\( blackbird, gingerroot \\), and \\( butterscotch \\) exist in the form \\( blackbird=-1-3\\,cheesecake,\\; gingerroot=-3-4\\,cheesecake,\\; butterscotch=-3-5\\,cheesecake \\). It now suffices to let \\( cheesecake=0 \\) and to note that \\( (1,5)=(4,-1)-(3,8)+2(0,7) \\) is in \\( caterpillar \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "m": "ceilingnumber",
        "n": "terminator",
        "n_1": "terminatorone",
        "g_1": "destroyerone",
        "g_2": "destroyertwo",
        "g_k": "destroyerkay",
        "u": "downward",
        "v": "horizontal",
        "w": "bigloser",
        "k": "constant",
        "a": "negative",
        "b": "voidvalue",
        "H": "superset"
      },
      "question": "B-1. In the additive group of ordered pairs of integers ( \\( ceilingnumber, terminator \\) ) [with addition defined componentwise: \\( \\left.(ceilingnumber, terminator)+\\left(ceilingnumber^{\\prime}, terminator^{\\prime}\\right)=\\left(ceilingnumber+ceilingnumber^{\\prime}, terminator+terminator^{\\prime}\\right)\\right] \\) consider the subgroup \\( superset \\) generated by the three elements\n\\[\n(3,8), \\quad(4,-1)\n\\]\n\nThen \\( \\boldsymbol{superset} \\) has another set of generators of the form\n\\[\n(1, voidvalue), \\quad(0, negative)\n\\]\nfor some integers \\( negative, voidvalue \\) with \\( negative>0 \\). Find \\( negative \\).\n[Elements \\( destroyerone, \\ldots, destroyerkay \\) are said to generate a subgroup \\( superset \\) if (i) each \\( destroyerone \\in superset \\), and (ii) every \\( h \\in superset \\) can be written as a sum \\( h=terminatorone destroyerone+\\cdots+n_{k} destroyerkay \\) where the \\( terminatorone \\) are integers (and where, for example, \\( 3 destroyerone-2 destroyertwo \\) means \\( \\left.\\left.destroyerone+destroyerone+destroyerone-destroyertwo-destroyertwo\\right).\\right] \\)",
      "solution": "B-1.\nThe answer is \\( negative=7 \\). Also one must have \\( voidvalue \\equiv 5(\\bmod 7) \\).\nProof: The subgroup \\( superset \\) must contain \\( 4(3,8)-3(4,-1)=(0,35), 4(5,4)-5(4,-1)=(0,21) \\), and then \\( 2(0,21)-(0,35)=(0,7) \\). Now \\( (0,7) \\) and \\( (1, voidvalue) \\) will generate \\( superset \\) iff \\( (1, voidvalue) \\) is in \\( superset \\) and there exist integers \\( downward \\), \\( horizontal \\), and \\( bigloser \\) such that\n\\[\n(3,8)=3(1, voidvalue)+downward(0,7),\\quad(4,-1)=4(1, voidvalue)+horizontal(0,7),\\quad(5,4)=5(1, voidvalue)+bigloser(0,7)\n\\]\n\nThese hold iff \\( 8=3 voidvalue+7 downward,\\,-1=4 voidvalue+7 horizontal \\), and \\( 4=5 voidvalue+7 bigloser \\). With \\( voidvalue=5+7 constant, constant \\) any integer, the desired coefficients \\( downward, horizontal \\), and \\( bigloser \\) exist in the form \\( downward=-1-3 constant, horizontal=-3-4 constant, bigloser=-3-5 constant \\). It now suffices to let \\( constant=0 \\) and to note that \\( (1,5)=(4,-1)-(3,8)+2(0,7) \\) is in \\( superset \\)."
    },
    "garbled_string": {
      "map": {
        "m": "qzxwvtnp",
        "n": "hjgrksla",
        "n_1": "zpyxvosk",
        "g_1": "pqowieuq",
        "g_2": "mznxbcva",
        "g_k": "cvbmalye",
        "u": "wienvckq",
        "v": "hqmsoder",
        "w": "zxclvmbn",
        "k": "qwertyui",
        "a": "asdfghjk",
        "b": "yxcvbnml",
        "H": "poiulkjh"
      },
      "question": "B-1. In the additive group of ordered pairs of integers ( \\( qzxwvtnp, hjgrksla \\) ) [with addition defined componentwise: \\( \\left.(qzxwvtnp, hjgrksla)+\\left(qzxwvtnp^{\\prime}, hjgrksla^{\\prime}\\right)=\\left(qzxwvtnp+qzxwvtnp^{\\prime}, hjgrksla+hjgrksla^{\\prime}\\right)\\right] \\) consider the subgroup \\( poiulkjh \\) generated by the three elements\n\\[\n(3,8), \\quad(4,-1)\n\\]\n\nThen \\( \\boldsymbol{poiulkjh} \\) has another set of generators of the form\n\\[\n(1, yxcvbnml), \\quad(0, asdfghjk)\n\\]\nfor some integers \\( asdfghjk, yxcvbnml \\) with \\( asdfghjk>0 \\). Find \\( asdfghjk \\).\n[Elements \\( pqowieuq, \\ldots, cvbmalye \\) are said to generate a subgroup \\( poiulkjh \\) if (i) each \\( pqowieuq \\in poiulkjh \\), and (ii) every \\( h \\in poiulkjh \\) can be written as a sum \\( h=zpyxvosk\\,pqowieuq+\\cdots+n_{k}\\,cvbmalye \\) where the \\( zpyxvosk \\) are integers (and where, for example, \\( 3\\,pqowieuq-2\\,mznxbcva \\) means \\( \\left.\\left.pqowieuq+pqowieuq+pqowieuq-mznxbcva-mznxbcva\\right).\\right] \\)",
      "solution": "B-1.\nThe answer is \\( asdfghjk=7 \\). Also one must have \\( yxcvbnml \\equiv 5(\\bmod 7) \\).\nProof: The subgroup \\( poiulkjh \\) must contain \\( 4(3,8)-3(4,-1)=(0,35), 4(5,4)-5(4,-1)=(0,21) \\), and then \\( 2(0,21)-(0,35)=(0,7) \\). Now \\( (0,7) \\) and \\( (1, yxcvbnml) \\) will generate \\( poiulkjh \\) iff \\( (1, yxcvbnml) \\) is in \\( poiulkjh \\) and there exist integers \\( wienvckq \\), \\( hqmsoder \\), and \\( zxclvmbn \\) such that\n\\[\n(3,8)=3(1, yxcvbnml)+wienvckq(0,7),\\quad(4,-1)=4(1, yxcvbnml)+hqmsoder(0,7),\\quad(5,4)=5(1, yxcvbnml)+zxclvmbn(0,7)\n\\]\n\nThese hold iff \\( 8=3\\,yxcvbnml+7\\,wienvckq,\\,-1=4\\,yxcvbnml+7\\,hqmsoder \\), and \\( 4=5\\,yxcvbnml+7\\,zxclvmbn \\). With \\( yxcvbnml=5+7\\,qwertyui,\\;qwertyui \\) any integer, the desired coefficients \\( wienvckq, hqmsoder \\), and \\( zxclvmbn \\) exist in the form \\( wienvckq=-1-3\\,qwertyui,\\;hqmsoder=-3-4\\,qwertyui,\\;zxclvmbn=-3-5\\,qwertyui \\). It now suffices to let \\( qwertyui=0 \\) and to note that \\( (1,5)=(4,-1)-(3,8)+2(0,7) \\) is in \\( poiulkjh \\)."
    },
    "kernel_variant": {
      "question": "In the additive group ℤ³ with component-wise addition, consider the subgroup  \n\n  H = ⟨ (4, 7, 9), (6, 2, 15), (8, 11, 3), (10, 5, 12) ⟩.  \n\nEvery full-rank sublattice of ℤ³ admits a Smith basis (1, 0, β), (0, 1, γ), (0, 0, α) with α > 0.  \nDetermine the value of α for this particular subgroup H.",
      "solution": "Note that cancelling the first coordinate is easy:  \n3(6,2,15) − 2(10,5,12) = (0, −4, −6), while 2(8,11,3) − (4,7,9) = (0, 15, −3).  \nSince both lie in H, their integer span contains every (0, y, z) obtained from linear combinations.  \nEliminating the second coordinate we take  \n15·(0, −4, −6)+4·(0, 15, −3) = (0, 0, −90).  \nA shorter element appears after lattice reduction:  \n5(4,7,9) − 10(6,2,15) − 5(8,11,3)+8(10,5,12) = (0, 0, −24).  \nHence the set S = { z ∈ ℤ : (0,0,z) ∈ H } contains ±24.  \n\nConversely, write 4a+6b+8c+10d = 0 and 7a+2b+11c+5d = 0.  \nThis gives b = −(6c+25d)/17 and, after substitution,  \nz = 9a+15b+3c+12d = 24(7d+11t) with t arbitrary.  \nBecause 7 and 11 are coprime, the parentheses run through every integer, so every z is divisible by 24 and 24 itself occurs; hence gcd S = 24.  \nTherefore S = 24ℤ and the last invariant factor equals 24.  \nHence H possesses a Smith basis (1,0,β), (0,1,γ), (0,0,24); in particular α = 24.  \n(Any companion vectors satisfy β ≡ 4, γ ≡ 7 (mod 24).) ∎",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.141905",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}