| { | |
| "log_file": [ | |
| "step_103-b_0.log", | |
| "step_103-b_1.log", | |
| "step_103-b_2.log", | |
| "step_103-b_3.log", | |
| "step_103-b_4.log", | |
| "step_103-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 three-digit numbers are multiples of neither 5 nor 7?\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 three-digit numbers are multiples of neither 5 nor 7?\n\n### Response:\nSEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/19.json\n\n#### Similar Question\nEvaluate $i^6+i^{16}+i^{-26}$.\n\n#### User Answer\nWe have $i^6 = i^4\\cdot i^2 = 1\\cdot (-1) = -1$. We also have $i^{16} = (i^4)^4 = 1^4 =1$, and $i^{-26} = 1/i^{26} = 1/(i^{24}\\cdot i^2) = 1/[1\\cdot (-1)] = -1$. So, adding these three results gives $i^6 + i^{16} + i^{-26} = -1+1-1 = -1$.\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 three-digit numbers are multiples of neither 5 nor 7?\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 three-digit numbers are multiples of neither 5 nor 7?\n\n### Response:\nSEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/210.json\n\n#### Similar Question\nLet $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?\n\n#### User Answer\nSuppose $f(z)=z^2+iz+1=c=a+bi$. We look for $z$ with $\\text{Im}(z)>0$ such that $a,b$ are integers where $|a|, |b|\\leq 10$.\n\nFirst, use the quadratic formula:\n\n$ z = \\frac{1}{2} (-i \\pm \\sqrt{-1-4(1-c)}) = -\\frac{i}{2} \\pm \\sqrt{ -\\frac{5}{4} + c }$\n\nGenerally, consider the imaginary part of a radical of a complex number: $\\sqrt{u}$, where $u = v+wi = r e^{i\\theta}$.\n\n$\\Im (\\sqrt{u}) = \\Im(\\pm \\sqrt{r} e^{i\\theta/2}) = \\pm \\sqrt{r} \\sin(i\\theta/2) = \\pm \\sqrt{r}\\sqrt{\\frac{1-\\cos\\theta}{2}} = \\pm \\sqrt{\\frac{r-v}{2}}$.\n\nNow let $u= -5/4 + c$, then $v = -5/4 + a$, $w=b$, $r=\\sqrt{v^2 + w^2}$.\n\nNote that $\\Im(z)>0$ if and only if $\\pm \\sqrt{\\frac{r-v}{2}}>\\frac{1}{2}$. The latter is true only when we take the positive sign, and that $r-v > 1/2$,\n\nor $v^2 + w^2 > (1/2 + v)^2 = 1/4 + v + v^2$, $w^2 > 1/4 + v$, or $b^2 > a-1$.\n\nIn other words, for all $z$, $f(z)=a+bi$ satisfies $b^2 > a-1$, and there is one and only one $z$ that makes it true. Therefore we are just going to count the number of ordered pairs $(a,b)$ such that $a$, $b$ are integers of magnitude no greater than $10$, and that $b^2 \\geq a$.\n\nWhen $a\\leq 0$, there is no restriction on $b$ so there are $11\\cdot 21 = 231$ pairs;\n\nwhen $a > 0$, there are $2(1+4+9+10+10+10+10+10+10+10)=2(84)=168$ pairs.\n\nThus there are $231+168=399$ numbers in total.\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 three-digit numbers are multiples of neither 5 nor 7?\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 three-digit numbers are multiples of neither 5 nor 7?\n\n### Response:\nSEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/number_theory/7075.json\n\n#### Similar Question\nFind the largest integer $n$ satisfying the following conditions:\n(i) $n^2$ can be expressed as the difference of two consecutive cubes;\n(ii) $2n + 79$ is a perfect square.\n\n#### User Answer\nWrite $n^2 = (m + 1)^3 - m^3 = 3m^2 + 3m + 1$, or equivalently, $(2n + 1)(2n - 1) = 4n^2 - 1 = 12m^2 + 12m + 3 = 3(2m + 1)^2$.\nSince $2n + 1$ and $2n - 1$ are both odd and their difference is $2$, they are relatively prime. But since their product is three times a square, one of them must be a square and the other three times a square. We cannot have $2n - 1$ be three times a square, for then $2n + 1$ would be a square congruent to $2$ modulo $3$, which is impossible.\nThus $2n - 1$ is a square, say $b^2$. But $2n + 79$ is also a square, say $a^2$. Then $(a + b)(a - b) = a^2 - b^2 = 80$. Since $a + b$ and $a - b$ have the same parity and their product is even, they are both even. To maximize $n$, it suffices to maximize $2b = (a + b) - (a - b)$ and check that this yields an integral value for $m$. This occurs when $a + b = 40$ and $a - b = 2$, that is, when $a = 21$ and $b = 19$. This yields $n = 181$ and $m = 104$, so the answer is $181$.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe first three-digit multiple of 5 is 100, and the first three-digit multiple of 7 is 700. Any number of the form $5k+3$ or $7k+4$ is a multiple of 5 or 7, so the first three-digit number that is a multiple of neither 5 nor 7 is $1000-3-4=\\boxed{993}$.</s>", | |
| " SEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]</s>", | |
| "The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nA number is a multiple of 5 if and only if it is a multiple of 25. A number is a multiple of 7 if and only if it is a multiple of 7. A number is a multiple of both 5 and 7 if and only if it is a multiple of 35. Therefore, the number of three-digit multiples of 5 or 7 is equal to the number of three-digit multiples of 25, 7, or 35. The three-digit multiples of 25 are 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, ... The three-digit multiples of 7 are 7, 14, 21, 28, 35, ... The three-digit multiples of 35 are 35, 70, 105, 140, ... The number of three-digit multiples of 5 or 7 is $1+4+6+8+10+12+14+...+48+50+52+...+98+100+102+104+106+108+110+112+114+116+118+120+122+124+126+128+130+132+134+136+138+140+142+144+146+148+150+152+154+156+158+160+162+164+166+168+170+172+174+176+178+180+182+184+186+188+190+192+194+196+198+200+202+204+206+208+210+212+214+216+218+220+222+224+226+228+230+232+234+236+238+240+242+244+246+248+250+252+254+256+258+2", | |
| " SEARCH[\"three-digit numbers\", \"$x^2 = -1$\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe three-digit multiples of 5 are 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295.\n\nThe three-digit multiples of 7 are 105, 112, 119, 132, 139, 146, 153, 160, 167, 174, 181, 188, 195, 202, 219, 226, 235, 242, 249, 256, 263, 270, 277, 284, 291, 298.\n\nThe three-digit multiples of 5 and 7 are 105, 140, 210, 280.\n\nThus there are $25 \\cdot 2 = 50$ three-digit multiples of 5 and $21 \\cdot 2 = 42$ three-digit multiples of 7. However, the multiples of 5 and 7 are disjoint, so we have counted some numbers twice. We must subtract the multiples of 35, which are 175, 245, and 320. Therefore, there are $50 + 42 - 3 = \\boxed{99}$ three-digit numbers which are multiples of neither 5 nor 7.</s>" | |
| ], | |
| "src_path": [ | |
| "train/counting_and_probability/174.json", | |
| "train/counting_and_probability/174.json", | |
| "train/counting_and_probability/174.json", | |
| "train/counting_and_probability/174.json", | |
| "train/counting_and_probability/174.json", | |
| "train/counting_and_probability/174.json" | |
| ], | |
| "input": [ | |
| "How many three-digit numbers are multiples of neither 5 nor 7?", | |
| "How many three-digit numbers are multiples of neither 5 nor 7?", | |
| "How many three-digit numbers are multiples of neither 5 nor 7?", | |
| "How many three-digit numbers are multiples of neither 5 nor 7?", | |
| "How many three-digit numbers are multiples of neither 5 nor 7?", | |
| "How many three-digit numbers are multiples of neither 5 nor 7?" | |
| ], | |
| "output": [ | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.", | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.", | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.", | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.", | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.", | |
| "It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15 \\times 7 = 105$, and the largest multiple of 7 that is a three-digit number is $142 \\times 7 = 994$. Therefore, there are $142-15+1 = 128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\\times 5 = 100$, and the largest multiple of 5 that is a three digit number is $199\\times 5 =995$. So there are $199-20+1=180$ multiples of 5.\n\nNow notice that we have counted some numbers twice: those multiples of $5\\times7=35$. The smallest multiple of 35 is $3\\times 35 = 105$, the largest multiple of 35 is $28\\times35 =980$. So there are $28-3+1=26$ multiples of 35.\n\nWe have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26 = 282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5." | |
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