| { | |
| "log_file": [ | |
| "step_136-b_0.log", | |
| "step_136-b_1.log", | |
| "step_136-b_2.log", | |
| "step_136-b_3.log", | |
| "step_136-b_4.log", | |
| "step_136-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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\n\n### Response:\nSEARCH[\"1-9 = \\\\cos(24,516)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1710.json\n\n#### Similar Question\nDefine a function $A(m, n)$ by \\[ A(m,n) = \\left\\{ \\begin{aligned} &n+1& \\text{ if } m = 0 \\\\ &A(m-1, 1) & \\text{ if } m > 0 \\text{ and } n = 0 \\\\ &A(m-1, A(m, n-1))&\\text{ if } m > 0 \\text{ and } n > 0. \\end{aligned} \\right.\\]Compute $A(2, 1).$\n\n#### User Answer\nWe iterate through the definition of $A(m, n).$ Each step below is marked either $(1),$ $(2),$ or $(3),$ corresponding to the three parts of the definition of $A(m, n)$: \\[\\begin{aligned} A(2, 1)&\\stackrel{(3)}{=} A(1, A(2, 0)) \\\\ &\\stackrel{(2)}{=} A(1, A(1, 1)) \\\\ &\\stackrel{(3)}{=} A(1, A(0, A(1, 0))) \\\\ &\\stackrel{(2)}{=} A(1, A(0, A(0, 1))) \\\\ &\\stackrel{(1)}{=} A(1, A(0, 2)) \\\\ &\\stackrel{(1)}{=} A(1, 3) \\\\ &\\stackrel{(3)}{=} A(0, A(1, 2)) \\\\ &\\stackrel{(3)}{=} A(0, A(0, A(1, 1))). \\end{aligned}\\]In the last few steps, we actually computed $A(1, 1) = 3,$ so we have \\[A(2, 1) = A(0, A(0, 3)) = A(0, 4) = 5\\]by applying $(1)$ twice.\n\n(Note: the function $A(m, n)$ is called the Ackermann function. Because of its deeply recursive definition, $A(m, n)$ grows extremely quickly. For example, other values of $A(m, n)$ include $A(3, 3) = 29$ and $A(4, 2) = 2^{65536} - 3,$ which has tens of thousands of digits in base ten!)\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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\n\n### Response:\nSEARCH[\"1-9 = \\\\cos(24,516)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/909.json\n\n#### Similar Question\nThe function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \\begin{align*} f(x,x) &=x, \\\\ f(x,y) &=f(y,x), \\quad \\text{and} \\\\ (x + y) f(x,y) &= yf(x,x + y). \\end{align*}Calculate $f(14,52)$.\n\n#### User Answer\nWe can rewrite the third equation as \\[f(x, x+y) = \\frac{x+y}{y} \\cdot f(x, y),\\]or, making the substitution $t = x+y,$ \\[f(x, t) = \\frac{t}{t-x} \\cdot f(x, t-x)\\]whenever $x < t.$ In particular, if $r \\neq 0$ is the remainder when $t$ is divided by $x,$ then repeatedly applying this relation, we have \\[\\begin{aligned} f(x, t) &= \\frac{t}{t-x} \\cdot f(x, t-x) \\\\ &= \\frac{t}{t-x} \\cdot \\frac{t-x}{t-2x} \\cdot f(x, t-2x) \\\\ &= \\dotsb \\\\ &= \\frac{t}{t-x} \\cdot \\frac{t-x}{t-2x} \\cdots \\frac{r+x}{r} \\cdot f(x, r) \\\\ &= \\frac{t}{r} \\cdot f(x, r) \\end{aligned}\\]since the product telescopes. Then we may compute $f(14, 52)$ as follows, swapping the two arguments of $f$ as necessary using the second equation: \\[\\begin{aligned} f(14, 52) &= \\frac{52}{10} \\cdot f(14, 10) \\\\ &= \\frac{52}{10} \\cdot \\frac{14}{4} \\cdot f(10, 4) \\\\ &= \\frac{52}{10} \\cdot \\frac{14}{4} \\cdot \\frac{10}{2} \\cdot f(4, 2)\\\\ &= \\frac{52}{10} \\cdot \\frac{14}{4} \\cdot \\frac{10}{2} \\cdot \\frac{4}{2} \\cdot f(2, 2) \\\\ &= \\frac{52}{\\cancel{10}} \\cdot \\frac{14}{\\cancel{4}} \\cdot \\frac{\\cancel{10}}{2} \\cdot \\frac{\\cancel{4}}{2} \\cdot 2 \\\\ &= 364. \\end{aligned}\\]\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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\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:\nHow many integers 1-9 are divisors of the five-digit number 24,516?\n\n### Response:\nSEARCH[\"1-9 = \\\\cos(24,516)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/25086.json\n\n#### Similar Question\nTwo circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?\n\n[asy]\nunitsize(0.2 cm);\n\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=4;\ndraw(Circle((0,0),7)); draw(Circle((0,0),10));\ndot((0,0)); dot((7,0)); dot((10,0)); dot((0,7)); dot((8,6));\ndraw((0,0)--(8,6));\nlabel(\"$S (0,k)$\",(0,7.5),W);\ndraw((13,0)--(0,0)--(0,13),Arrows(TeXHead));\ndraw((-13,0)--(0,0)--(0,-13));\n\nlabel(\"$x$\",(13,0),E); label(\"$y$\",(0,13),N); label(\"$P(8,6)$\",(8,6),NE);\n\nlabel(\"$O$\",(0,0),SW); label(\"$Q$\",(7,0),SW); label(\"$R$\",(10,0),SE);\n\n[/asy]\n\n#### User Answer\nWe can determine the distance from $O$ to $P$ by dropping a perpendicular from $P$ to $T$ on the $x$-axis. [asy]\nunitsize(0.2 cm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=4;\ndraw(Circle((0,0),7)); draw(Circle((0,0),10));\ndot((0,0)); dot((7,0)); dot((10,0)); dot((0,7)); dot((8,6));\ndraw((0,0)--(8,6)--(8,0));\nlabel(\"$S (0,k)$\",(0,7.5),W);\ndraw((13,0)--(0,0)--(0,13),Arrows(TeXHead));\ndraw((-13,0)--(0,0)--(0,-13));\ndraw((8.8,0)--(8.8,.8)--(8,.8));\nlabel(\"$x$\",(13,0),E); label(\"$y$\",(0,13),N); label(\"$P(8,6)$\",(8,6),NE);\n\nlabel(\"$O$\",(0,0),SW); label(\"$Q$\",(7,0),SW); label(\"$T$\",(8,0),S); label(\"$R$\",(10,0),SE);\n\n[/asy] We have $OT=8$ and $PT=6$, so by the Pythagorean Theorem, \\[ OP^2 = OT^2 + PT^2 = 8^2+6^2=64+36=100 \\]Since $OP>0$, then $OP = \\sqrt{100}=10$. Therefore, the radius of the larger circle is $10$. Thus, $OR=10$.\n\nSince $QR=3$, then $OQ = OR - QR = 10 - 3 = 7$. Therefore, the radius of the smaller circle is $7$.\n\nSince $S$ is on the positive $y$-axis and is 7 units from the origin, then the coordinates of $S$ are $(0,7)$, which means that $k=7$.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"1-9 = \\\\cos(24,516)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe can factor the number 24,516 as $2^3\\cdot5\\cdot37$. We can see that the only integers 1-9 which divide 24,516 are 1, 2, 3, 5, 10, 15, 20, 30, and 35. So there are $\\boxed{9}$ such integers.</s>", | |
| " SEARCH[\"1-9 = \\\\cos(24,516)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe note that 24,516 can be written as $2^4 \\cdot 3^3 \\cdot 5$. Therefore, the divisors of 24,516 are the divisors of $2^4 \\cdot 3^3 \\cdot 5$, which are the divisors of $2^4 \\cdot 3^3$. We can find these divisors by writing $2^4 \\cdot 3^3$ as $2^4 \\cdot 3^2 \\cdot 3$. The divisors of this integer are then $2^0 \\cdot 3^0$, $2^1 \\cdot 3^0$, $2^2 \\cdot 3^0$, $2^1 \\cdot 3^1$, $2^2 \\cdot 3^1$, $2^3 \\cdot 3^0$, $2^2 \\cdot 3^2$, $2^3 \\cdot 3^1$, and $2^3 \\cdot 3^2$. We see that there are $\\boxed{9}$ integers in this list.</s>", | |
| " SEARCH[\"1-9 = \\\\cos(24,516)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe can simply list all the factors of 24,516. They are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 576, 1152, and 2304. Of these, the integers from 1 to 9 are 1, 2, 3, 4, 6, 8, 12, 24, and 36. So the answer is $\\boxed{9}$.</s>" | |
| ], | |
| "src_path": [ | |
| "train/number_theory/1127.json", | |
| "train/number_theory/1127.json", | |
| "train/number_theory/1127.json", | |
| "train/number_theory/1127.json", | |
| "train/number_theory/1127.json", | |
| "train/number_theory/1127.json" | |
| ], | |
| "input": [ | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?", | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?", | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?", | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?", | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?", | |
| "How many integers 1-9 are divisors of the five-digit number 24,516?" | |
| ], | |
| "output": [ | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9.", | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9.", | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9.", | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9.", | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9.", | |
| "We know that 24,516 is divisible by 1. Since 24,516 is even, it is also divisible by 2. The sum of the digits of 24,516 is $2+4+5+1+6=18$. A number is divisible by 3 if the sum of its digits is divisible by 3, so 24,516 is divisible by 3. For a number to be divisible by 4, its last two digits must be divisible by 4. Because 16 is divisible by 4, so is 24,516. 24,516 is not divisible by 5 because it does not end in a 5 or 0. It is divisible by 6 because it was divisible by 2 and 3. In order to see if a number is divisible by 7, it is necessary to double the last digit and subtract that value from the original number without the units digit. (In this case, the original number without the units digit is 2451.) If the resulting number is divisible by 7, then so is the original number. When 12 is subtracted from 2451, we get $2451-12=2439$. Since it is still not clear whether this number is divisible by 7, we repeat the process: $243-18=225$ and $22-10=12$. We can now see that 24,516 is not divisible by 7. In order for a number to be divisible by 8, its last three digits must be divisible by 8. Since 516 is not divisible by 8, neither is 24,516. Because the sum of the digits in 24,516 is divisible by 9, 24,516 is divisible by 9. We conclude that 24,516 is divisible by $\\boxed{6}$ of the integers from 1 to 9." | |
| ], | |
| "rewards": [ | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0 | |
| ] | |
| } |