logs/step_1000-b_0.log

{
  "log_file": [
    "step_1000-b_0.log",
    "step_1000-b_1.log",
    "step_1000-b_2.log",
    "step_1000-b_3.log",
    "step_1000-b_4.log",
    "step_1000-b_5.log"
  ],
  "query": [
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\nSEARCH[\"$|6x^2-47x+15|$ prime\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/8.json\n\n#### Similar Question\nGiven vectors $\\mathbf{a}$ and $\\mathbf{b},$ let $\\mathbf{p}$ be a vector such that\n\\[\\|\\mathbf{p} - \\mathbf{b}\\| = 2 \\|\\mathbf{p} - \\mathbf{a}\\|.\\]Among all such vectors $\\mathbf{p},$ there exists constants $t$ and $u$ such that $\\mathbf{p}$ is at a fixed distance from $t \\mathbf{a} + u \\mathbf{b}.$  Enter the ordered pair $(t,u).$\n\n#### User Answer\nFrom $\\|\\mathbf{p} - \\mathbf{b}\\| = 2 \\|\\mathbf{p} - \\mathbf{a}\\|,$\n\\[\\|\\mathbf{p} - \\mathbf{b}\\|^2 = 4 \\|\\mathbf{p} - \\mathbf{a}\\|^2.\\]This expands as\n\\[\\|\\mathbf{p}\\|^2 - 2 \\mathbf{b} \\cdot \\mathbf{p} + \\|\\mathbf{b}\\|^2 = 4 \\|\\mathbf{p}\\|^2 - 8 \\mathbf{a} \\cdot \\mathbf{p} + 4 \\|\\mathbf{a}\\|^2,\\]which simplifies to $3 \\|\\mathbf{p}\\|^2 = 8 \\mathbf{a} \\cdot \\mathbf{p} - 2 \\mathbf{b} \\cdot \\mathbf{p} - 4 \\|\\mathbf{a}\\|^2 + \\|\\mathbf{b}\\|^2.$  Hence,\n\\[\\|\\mathbf{p}\\|^2 = \\frac{8}{3} \\mathbf{a} \\cdot \\mathbf{p} - \\frac{2}{3} \\mathbf{b} \\cdot \\mathbf{p} - \\frac{4}{3} \\|\\mathbf{a}\\|^2 + \\frac{1}{3} \\|\\mathbf{b}\\|^2.\\]We want $\\|\\mathbf{p} - (t \\mathbf{a} + u \\mathbf{b})\\|$ to be constant, which means $\\|\\mathbf{p} - t \\mathbf{a} - u \\mathbf{b}\\|^2$ is constant.  This expands as\n\\begin{align*}\n\\|\\mathbf{p} - t \\mathbf{a} - u \\mathbf{b}\\|^2 &= \\|\\mathbf{p}\\|^2 + t^2 \\|\\mathbf{a}\\|^2 + u^2 \\|\\mathbf{b}\\|^2 - 2t \\mathbf{a} \\cdot \\mathbf{p} - 2u \\mathbf{b} \\cdot \\mathbf{p} + 2tu \\mathbf{a} \\cdot \\mathbf{b} \\\\\n&= \\frac{8}{3} \\mathbf{a} \\cdot \\mathbf{p} - \\frac{2}{3} \\mathbf{b} \\cdot \\mathbf{p} - \\frac{4}{3} \\|\\mathbf{a}\\|^2 + \\frac{1}{3} \\|\\mathbf{b}\\|^2 \\\\\n&\\quad + t^2 \\|\\mathbf{a}\\|^2 + u^2 \\|\\mathbf{b}\\|^2 - 2t \\mathbf{a} \\cdot \\mathbf{p} - 2u \\mathbf{b} \\cdot \\mathbf{p} + 2tu \\mathbf{a} \\cdot \\mathbf{b} \\\\\n&= \\left( \\frac{8}{3} - 2t \\right) \\mathbf{a} \\cdot \\mathbf{p} - \\left( \\frac{2}{3} + 2u \\right) \\mathbf{b} \\cdot \\mathbf{p} \\\\\n&\\quad + \\left( t^2 - \\frac{4}{3} \\right) \\|\\mathbf{a}\\|^2 + \\left( u^2 + \\frac{1}{3} \\right) \\|\\mathbf{b}\\|^2 + 2tu \\mathbf{a} \\cdot \\mathbf{b}.\n\\end{align*}The only non-constant terms in this expression are $\\left( \\frac{8}{3} - 2t \\right) \\mathbf{a} ...\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\nSEARCH[\"$|6x^2-47x+15|$ prime\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1202.json\n\n#### Similar Question\nLet $x,$ $y,$ $z,$ $v,$ $w$ be positive real numbers such that $x^2 + y^2 + z^2 + v^2 + w^2 = 2016.$  Let $M$ be the maximum value of\n\\[xz + 2yz + 3zv + 7zw,\\]and let $x_M,$ $y_M$, $z_M,$ $v_M,$ $w_M$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ respectively, that produce the maximum value of $M.$  Find $M + x_M + y_M + z_M + v_M + w_M.$\n\n#### User Answer\nNote that $xz + 2yz + 3zv + 7zw = z(x + 2y + 3v + 7w).$  By Cauchy-Schwarz,\n\\begin{align*}\nx + 2y + 3v + 7w &\\le \\sqrt{(1 + 4 + 9 + 49)(x^2 + y^2 + v^2 + w^2)} \\\\\n&= \\sqrt{63 (x^2 + y^2 + v^2 + w^2)} \\\\\n&= 3 \\sqrt{7(2016 - z^2)},\n\\end{align*}so $z(x + 2y + 3v + 7w) \\le 3z \\sqrt{7(2016 - z^2)} = 3 \\sqrt{7z^2 (2016 - z^2)}.$\n\nBy AM-GM,\n\\[z^2 (2016 - z^2) \\le \\left( \\frac{z^2 + (2016 - z^2)}{2} \\right)^2 = 1008^2,\\]so\n\\[3 \\sqrt{7z^2 (2016 - z^2)} \\le 3 \\sqrt{7 \\cdot 1008^2} = 3024 \\sqrt{7}.\\]Equality occurs when $x:y:v:w = 1:2:3:7,$ $z^2 = 1008,$ and $x^2 + y^2 + z^2 + v^2 + w^2 = 2016,$ which leads to $x = 4,$ $y = 8,$ $z = 12 \\sqrt{7},$ $v = 12$, and $w = 28.$  Thus,\n\\[M + x_M + y_M + z_M + v_M + w_M = 3024 \\sqrt{7} + 4 + 8 + 12 \\sqrt{7} + 12 + 28 = 52 + 3036 \\sqrt{7}.\\]\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?\n\n### Response:\nSEARCH[\"$|6x^2-47x+15|$ prime\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1816.json\n\n#### Similar Question\nLet $a,$ $b,$ $c$ be distinct integers, and let $\\omega$ be a complex number such that $\\omega^3 = 1$ and $\\omega \\neq 1.$  Find the smallest possible value of\n\\[|a + b \\omega + c \\omega^2|.\\]\n\n#### User Answer\nNote that $|\\omega^3| = |\\omega|^3 = 1,$ so $|\\omega| = 1.$   Then $\\omega \\overline{\\omega} = |\\omega|^2 = 1.$\n\nAlso, $\\omega^3 - 1 = 0,$ which factors as $(\\omega - 1)(\\omega^2 + \\omega + 1) = 0.$  Since $\\omega \\neq 1,$\n\\[\\omega^2 + \\omega + 1 = 0.\\]Hence,\n\\begin{align*}\n|a + b \\omega + c \\omega^2|^2 &= (a + b \\omega + c \\omega^2)(a + b \\overline{\\omega} + c \\overline{\\omega^2}) \\\\\n&= (a + b \\omega + c \\omega^2) \\left( a + \\frac{b}{\\omega} + \\frac{c}{\\omega^2} \\right) \\\\\n&= (a + b \\omega + c \\omega^2)(a + b \\omega^2 + c \\omega) \\\\\n&= a^2 + b^2 + c^2 + (\\omega + \\omega^2) ab + (\\omega + \\omega^2) ac + (\\omega^2 + \\omega^4) bc \\\\\n&= a^2 + b^2 + c^2 + (\\omega + \\omega^2) ab + (\\omega + \\omega^2) ac + (\\omega + \\omega^2) bc \\\\\n&= a^2 + b^2 + c^2 - ab - ac - bc \\\\\n&= \\frac{(a - b)^2 + (a - c)^2 + (b - c)^2}{2}.\n\\end{align*}Since $a,$ $b,$ and $c$ are distinct, all three of $|a - b|,$ $|a - c|,$ and $|b - c|$ must be at least 1, and at least one of these absolute values must be at least 2, so\n\\[\\frac{(a - b)^2 + (a - c)^2 + (b - c)^2}{2} \\ge \\frac{1 + 1 + 4}{2} = 3.\\]Equality occurs when $a,$ $b,$ and $c$ are any three consecutive integers, in any order, so the smallest possible value of $|a + b \\omega + c \\omega^2|$ is $\\sqrt{3}.$\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"$|6x^2-47x+15|$ prime\"]</s>",
    "The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe know that $|6x^2-47x+15|$ is prime only when $6x^2-47x+15$ is prime.  The prime factors of $6x^2-47x+15$ are $3$ and $5$, and the only way that $6x^2-47x+15$ can be prime is if $3$ and $5$ are both multiplied together.  Since there is only one $3$ and one $5$ in the expression, we know that $6x^2-47x+15$ is prime only when $3\\cdot 5 = 15$ is multiplied together, and only when $6x^2-47x+15 = 3 \\cdot 5 \\cdot (2x-3) = 3 \\cdot 5 \\cdot 2x - 3 \\cdot 5 = 10x-15$ is multiplied together.  Since we know that $6x^2-47x+15$ is prime only when $10x-15$ is multiplied together, we know that $6x^2-47x+15$ is prime only when $10x-15 = 3 \\cdot 5 \\cdot 2x - 3 \\cdot 5$ is multiplied together.  Since we know that $10x-15$ is prime only when $2x-3$ is multiplied together, we know that $6x^2-47x+15$ is prime only when $2x-3 = 3 \\cdot 5 \\cdot ",
    " SEARCH[\"$|6x^2-47x+15|$ prime\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nBecause we are looking for the greatest integer $x$, we can first assume that $6x^2 - 47x + 15$ is not equal to 0. If it were equal to 0, then we would have $6x^2 - 47x + 15 = 0$. Factoring, we get $(2x - 3)(3x - 5) = 0$. Thus, $x = \\frac{3}{2}$ or $x = \\frac{5}{3}$. However, because we are looking for the greatest integer $x$, we can only use $x = \\boxed{3}$.</s>",
    " SEARCH[\"$|6x^2-47x+15|$ prime\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nNote that $|6x^2-47x+15| = 6|x^2-7x+25|$. We know that $x^2-7x+25 = (x-5)^2$ is always nonnegative, so $|6x^2-47x+15| = 6\\cdot 0 = 0$ when $x = 5$.  Otherwise, $|6x^2-47x+15| = 6|x-5|$, so the greatest integer $x$ such that $|6x^2-47x+15|$ is prime is $\\boxed{5}$.</s>"
  ],
  "src_path": [
    "train/algebra/475.json",
    "train/algebra/475.json",
    "train/algebra/475.json",
    "train/algebra/475.json",
    "train/algebra/475.json",
    "train/algebra/475.json"
  ],
  "input": [
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?",
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?",
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?",
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?",
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?",
    "What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime?"
  ],
  "output": [
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime.",
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime.",
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime.",
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime.",
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime.",
    "We can factor the quadratic $6x^2-47x+15$ as $(2x-15)(3x-1)$. So we have $|6x^2-47x+15|=|(2x-15)(3x-1)|=|2x-15|\\cdot|3x-1|$. In order for $|6x^2-47x+15|$ to be prime, its only divisors must be $1$ and itself. Thus one of $|2x-15|$ or $|3x-1|$ must be equal to $1$.\n\nIf $|3x-1|=1$, then $3x-1=1$ or $3x-1=-1$. These equations yield $x=\\frac{2}{3}$ and $x=0$, respectively. We throw out $x=\\frac{2}{3}$ since it is not an integer, and keep $x=0$ as a candidate.\n\nIf $|2x-15|=1$, then $2x-15=1$, in which case $2x=16$ and $x=8$, or $2x-15=-1$, in which case $2x=14$ and $x=7$.\n\nThus our candidates for the greatest $x$ are $0, 7$, and $8$. It remains to check whether the other factor is prime. We first check $x=8$. Since $|2x-15|=1$, we know $|2x-15|\\cdot|3x-1|=|3x-1|=|24-1|=23$, which is prime. Thus $\\boxed{8}$ is the largest integer for which $|6x^2-47x+15|$ is prime."
  ],
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    0.0,
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}