{ "log_file": [ "step_12-b_0.log", "step_12-b_1.log", "step_12-b_2.log", "step_12-b_3.log", "step_12-b_4.log", "step_12-b_5.log" ], "query": [ "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\n", "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\nSEARCH[\"greatest\", \"$formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1033.json\n\n#### Similar Question\nThe roots of the equation $x^2+kx+5 = 0$ differ by $\\sqrt{61}$. Find the greatest possible value of $k$.\n\n#### User Answer\nBy the quadratic formula, the roots of the equation are \\begin{align*}\n\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}&=\\frac{-k\\pm\\sqrt{k^2-4(5)(1)}}{2(1)}\\\\\n&=\\frac{-k\\pm\\sqrt{k^2-20}}{2}.\n\\end{align*} We want the difference of the roots, so we take the larger minus the smaller: \\begin{align*}\n\\left(\\frac{-k+\\sqrt{k^2-20}}{2}\\right)-\\left(\\frac{-k-\\sqrt{k^2-20}}{2}\\right)&=\\frac{2\\sqrt{k^2-20}}{2}\\\\\n&=\\sqrt{k^2-20}.\n\\end{align*} We are given that this difference is equal to $\\sqrt{61}$, so we have \\begin{align*}\n\\sqrt{k^2-20}&=\\sqrt{61}\\quad\\Rightarrow\\\\\nk^2-20&=61\\quad\\Rightarrow\\\\\nk^2&=81\\quad\\Rightarrow\\\\\nk&=\\pm 9.\n\\end{align*} Thus the greatest possible value of $k$ is $9$.\n\n--- RESULTS END ---\n\n", "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\n", "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\nSEARCH[\"greatest\", \"$formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/70.json\n\n#### Similar Question\nEvaluate $\\lfloor17.2\\rfloor+\\lfloor-17.2\\rfloor$.\n\n#### User Answer\nThe greatest integer less than $17.2$ is $17$, and the greatest integer less than $-17.2$ is $-18$, so our answer is $-1$.\n\n--- RESULTS END ---\n\n", "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\n", "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:\nLet $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n\n\n### Response:\nSEARCH[\"greatest\", \"$formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/number_theory/1099.json\n\n#### Similar Question\nGiven that $x$ is a multiple of $15336$, what is the greatest common divisor of $f(x)=(3x+4)(7x+1)(13x+6)(2x+9)$ and $x$?\n\n#### User Answer\nIn $f(x)$, all terms will have a multiple of $x$ except for the constant term, which is the multiple of the four constants $4,1,6$, and $9$.\n\nRecall (from the Euclidean algorithm) that the greatest common divisor of $a$ and $b$ is the same as the greatest common divisor of $a$ and $a-kb$ where $k,a,$ and $b$ are any integers. Therefore, finding the greatest common divisor of $f(x)$ and $x$ is the same as finding the greatest common divisor of $x$ and the constant term of $f(x)$. Therefore, we want to find \\begin{align*}\n\\text{gcd}\\,((3x+4)(7x+1)(13x+6)(2x+9),x) &=\\text{gcd}\\,(4 \\cdot 1 \\cdot 6 \\cdot 9, x)\\\\\n&=\\text{gcd}\\,(216,x)\n\\end{align*}Since $15336$ is a multiple of $216$, the greatest common divisor of $f(x)$ and $x$ is $216$.\n\n--- RESULTS END ---\n\n" ], "response": [ " SEARCH[\"greatest\", \"$formula\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe can test numbers in reverse order, starting with 999. If we subtract 7 from 999, we get 992, which has all distinct digits. If we subtract 7 from 992, we get 985, which has all distinct digits. If we subtract 7 from 985, we get 978, which has all distinct digits. If we subtract 7 from 978, we get 971, which has all distinct digits. If we subtract 7 from 971, we get 964, which has all distinct digits. If we subtract 7 from 964, we get 957, which has all distinct digits. If we subtract 7 from 957, we get 949, which has all distinct digits. If we subtract 7 from 949, we get 942, which has all distinct digits. If we subtract 7 from 942, we get 935, which has all distinct digits. If we subtract 7 from 935, we get 928, which has all distinct digits. If we subtract 7 from 928, we get 921, which has all distinct digits. If we subtract 7 from 921, we get 914, which has all distinct digits. If we subtract 7 from 914, we get 907, which has all distinct digits. If we subtract 7 from 907, we get 900, which has all distinct digits. Thus, the greatest integer multiple of 8 whose digits are all distinct is 900, and the remainder when 900 is divided by 1000 is $\\boxed{100}$.", " SEARCH[\"greatest\", \"$formula\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\nNote that any integer of the form $8k$ or $8k+1$ will fail to have all digits distinct, since $8$ and $9$ will always appear together. Therefore, we need to look only at integers of the form $8k+2$.\nWe can easily see that $8\\cdot1+2=10$ and $8\\cdot2+2=18$ are both possible (and have distinct digits). Therefore, we need only look at integers of the form $8\\cdot3+2=26$, $8\\cdot4+2=38$, and $8\\cdot5+2=48$. We can see that $26$ and $38$ are possible (and have distinct digits), but $48$ is not, since it is divisible by $2$ and $4$. Therefore, the only possible integers are $26$ and $38$, and the remainder when $N$ is divided by $1000$ is $\\boxed{26}$.", " SEARCH[\"greatest\", \"$formula\"]", "The result might be helpful, I will try using it to answer the question only if it is useful.\n\nWe can start by listing out the first few multiples of 8.\n\n$$8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, \\ldots$$\n\nWe can see that the first few multiples of 8 have digits that repeat. However, as we go further, the digits start to change more frequently. For example, 88 has a digit that is not 0, 8, or 2. 96 has a digit that is not 6. 104 has a digit that is not 4. 112 has a digit that is not 2. 120 has a digit that is not 0. 128 has a digit that is not 8. 136 has a digit that is not 6. 144 has a digit that is not 4. 152 has a digit that is not 2. 160 has a digit that is not 0. 168 has a digit that is not 8. 176 has a digit that is not 6. 184 has a digit that is not 4. 192 has a digit that is not 2. 200 has a digit that is not 0. 208 has a digit that is not 8. 216 has a digit that is not 6. 224 has a digit that is not 4. 232 has a digit that is not 2. 240 has a digit that is not 0. 248 has a digit that is not 8. 256 has a digit that is not 6. 264 has a digit that is not 4. 272 has a digit that is not 2. 280 has a digit that is not 0. 288 has a digit that is not 8. 296 has a digit that is not 6. 304 has a digit that is not 4. 312 has a digit that is not 2. 320 has a digit that is not 0. 328 has a digit that is not 8. 336 has a digit that is not 6. 344 has a digit that is not 4. 352 has a digit that is not 2. 360 has a digit that is not 0. 368 has a digit that is not 8. 376 has a digit that is not 6. 384 has a digit that is not 4. 392 has a digit that is not 2. 400 has a digit that is not 0. 408 has a digit that is not 8. 416 has a digit that is not 6. 424 has a digit that is not 4. 432 has a digit that is not 2. 440 has a digit that is not 0. 448 has a digit that is not 8. 456 has a digit that is not 6. 464 has a digit that is not 4. 472 has a digit that is not 2. 480 has a digit that is not 0. 488 has a digit that is not 8. 496 has a digit that is not 6. 504 has a digit that is not 4. 512 has a digit that is not 2. 520 has a digit that is not 0. 528 has a digit that is not 8. 536 has a digit that is not 6. 544 has a digit that is not 4. 552 has a digit that is not 2. 560 has a digit that is not 0. 568 has a digit that is not 8. 576 has a digit that is not 6. 584 has a digit that is" ], "src_path": [ "train/number_theory/7049.json", "train/number_theory/7049.json", "train/number_theory/7049.json", "train/number_theory/7049.json", "train/number_theory/7049.json", "train/number_theory/7049.json" ], "input": [ "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n", "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n", "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n", "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n", "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n", "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?\n" ], "output": [ "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$.", "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$.", "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$.", "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$.", "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$.", "We want a number with no digits repeating, so we can only use the digits 0-9 once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last three digits of the greatest number should be an arrangement of the digits $0,1,2$. Since the number has to be divisible by 8, the integer formed by the arrangement of $0,1,2$ is also divisible by 8. The only arrangement that works is $120$.\nTherefore, the remainder when the number is divided by $1000$ is $\\boxed{120}$." ], "rewards": [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ] }