| { | |
| "log_file": [ | |
| "step_112-b_0.log", | |
| "step_112-b_1.log", | |
| "step_112-b_2.log", | |
| "step_112-b_3.log", | |
| "step_112-b_4.log", | |
| "step_112-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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\n\n### Response:\nSEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/number_theory/823.json\n\n#### Similar Question\nKaleb defined a $\\emph{clever integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 9. What fraction of all clever integers is divisible by 27? Express your answer as a common fraction.\n\n#### User Answer\nYou can easily list out the numbers between 20 and 120 that have digits that add to 9: 27, 36, 45, 54, 63, 72, 81, 90, 108, 117.\n\nTo be a $\\emph{clever integer}$ the number must be even, which leaves us with 36, 54, 72, 90, and 108, a total of 5 numbers.\n\nThe problem asks which fraction of these five clever integers are divisible by 27. Only two, 54 and 108 are divisible by 27, thus the total fraction is $\\frac{2}{5}$.\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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\n\n### Response:\nSEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/911.json\n\n#### Similar Question\nHow many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?\n\n#### User Answer\nSince the boxes are indistinguishable, there are 5 different cases for arrangements of the number of balls in each box: $(5,0,0)$, $(4,1,0)$, $(3,2,0)$, $(3,1,1)$, or $(2,2,1)$.\n\n$(5,0,0)$: There is only $1$ way to put all 5 balls in one box.\n\n$(4,1,0)$: There are $\\binom{5}{4} = 5$ choices for the 4 balls in one of the boxes.\n\n$(3,2,0)$: There are $\\binom{5}{3} = 10$ choices for the 3 balls in one of the boxes.\n\n$(3,1,1)$: There are $\\binom{5}{3} = 10$ choices for the 3 balls in one of the boxes, and we simply split the last two among the other indistinguishable boxes.\n\n$(2,2,1)$: There are $\\binom{5}{2} = 10$ options for one of the boxes with two balls, then $\\binom{3}{2} = 3$ options for the second box with two balls, and one option remaining for the third. However since the boxes with two balls are indistinguishable, we are counting each pair of balls twice, and must divide by two. So there are $\\dfrac{10 \\times 3}{2} = 15$ arrangements of balls as $(2,2,1)$.\n\nThus the total number of arrangements for 3 indistinguishable boxes and 5 distinguishable balls is $1 + 5 + 10 + 10 + 15 = 41$.\n\n$\\textbf{Alternate solution:}$ There are $3^5 = 243$ arrangements to put 5 distinguishable balls in 3 distinguishable boxes. Among these 243 arrangements, there is one case in our problem that is counted three times: if all 5 balls are placed in one box and the other two boxes both contain nothing. This leaves 240 other arrangements.\n\nFor every other case, the contents of each box is different, and so these cases are each counted $3! = 6$ times. Therefore there must be 40 of these cases, and we have $41$ cases 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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\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 positive three-digit integers with a $5$ in the units place are divisible by $15$?\n\n### Response:\nSEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/12.json\n\n#### Similar Question\nAfter traveling 50 miles by taxi, Ann is charged a fare of $\\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be charged (in dollars) if she had traveled 70 miles?\n\n#### User Answer\nLet $d$ equal the cost of a 70 mile taxi ride. Since we know that Ann was charged $120 dollars a 50 mile taxi ride, we can set up the proportion $\\frac{120}{50}=\\frac{d}{70}$. If we solve for $d$ by multiplying both sides by 70, we find that $d=\\left(\\frac{120}{50}\\right)(70)=168$ dollars.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nAn integer is divisible by $15$ if and only if it is divisible by both $3$ and $5$. An integer that ends in $5$ is divisible by $5$, so we only need to consider whether or not the integer is divisible by $3$. An integer that ends in $5$ and is divisible by $3$ must end in $0$ or $5$. Since we are looking for three-digit integers, the only possibility is that the integer is $500+5(100+x)=500+500+5x$, where $x$ is a digit. We see that $5x$ can range from $0$ to $9$, so there are $10$ possible integers. However, $500+500+500=1000$ is not a three-digit integer, so there are only $\\boxed{9}$ possible integers.</s>", | |
| " SEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nA number $a$ is divisible by $15$ if and only if it is divisible by both $3$ and $5$. A number $a$ is divisible by $3$ if and only if the sum of its digits is divisible by $3$. A number $a$ is divisible by $5$ if and only if its units digit is a $0$ or a $5$. Therefore, a number $a$ is divisible by both $3$ and $5$ if and only if its units digit is a $0$ or a $5$.\n\nThere are $9$ three-digit numbers with a units digit of $0$ or $5$. Of these, $5$ end in $0$, $3$ end in $5$, and $1$ ends in $1$. Therefore, there are $9-5-3-1=\\boxed{0}$ three-digit numbers with a units digit of $0$ or $5$ that are divisible by $15$.</s>", | |
| " SEARCH[\"$5$ in the units place\", \"positive three-digit integers\", \"divisible by $15$\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nAn integer is divisible by $15$ if and only if it is divisible by both $3$ and $5$.\n\nAn integer in the form $5x+y$ is divisible by $3$ if and only if the integer in the form $x+y$ is divisible by $3$. We see that this is true if and only if $x+y$ is divisible by $3$. Since $0\\leq y\\leq 9$, we see that there are $\\boxed{10}$ possible values for $x$ that will make $5x+y$ divisible by $3$, and thus, $5x+y$ will be divisible by $5$.\n\nThe answer is 10</s>" | |
| ], | |
| "src_path": [ | |
| "train/number_theory/262.json", | |
| "train/number_theory/262.json", | |
| "train/number_theory/262.json", | |
| "train/number_theory/262.json", | |
| "train/number_theory/262.json", | |
| "train/number_theory/262.json" | |
| ], | |
| "input": [ | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?", | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?", | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?", | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?", | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?", | |
| "How many positive three-digit integers with a $5$ in the units place are divisible by $15$?" | |
| ], | |
| "output": [ | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15.", | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15.", | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15.", | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15.", | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15.", | |
| "Odd multiples of 5 have a 5 in the units place, while even multiples of 5 have a 0 in the units place. Multiples of 15 are multiples of 5, so we look at how many three-digit multiples of 15 are odd to find how many have a 5 in the units place. The three-digit multiples of 15 range from 105 to 990, or $15\\times7$ to $15\\times 66$. So there are $66-7+1=60$ three-digit multiples of 15. Half of them will be odd, so there are 30 odd, three-digit multiples of 15. There are $\\boxed{30}$ positive, three-digit integers with a 5 in the units place that are divisible by 15." | |
| ], | |
| "rewards": [ | |
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