| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the greatest common divisor (GCD) of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\), we need to follow these steps: 1. **Identify the integers in the set**: The set contains the integers \(1187\), \(-\frac{5}{2}\), and \(-\frac{1247}{3}\). Since the GCD is defined for integers, we need to convert the fractions to integers. 2. **Convert fractions to integers**: - \(-\frac{5}{2}\) is already in fractional form. - \(-\frac{1247}{3}\) is already in fractional form. 3. **Determine the GCD of the integers**: - The integer \(1187\) is a prime number. - The fractions \(-\frac{5}{2}\) and \(-\frac{1247}{3}\) are not integers, so they do not contribute to the GCD calculation directly. 4. **Conclusion**: - Since \(1187\) is the only integer in the set and it is prime, its GCD with any other number (including fractions) is 1. Therefore, the greatest common divisor of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\) is \(\boxed{1}\). |
\frac{1}{6} |
| deepseek-r1 | To find the greatest common divisor (GCD) of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\), we need to consider the GCD of the numerators and the least common multiple (LCM) of the denominators when the numbers are expressed as fractions. 1. **Convert all numbers to fractions:** - \(1187 = \frac{1187}{1}\) - \(-\frac{5}{2}\) - \(-\frac{1247}{3}\) 2. **Find the GCD of the numerators (1187, 5, 1247):** - Using the Euclidean algorithm: - GCD(1187, 5): - \(1187 \div 5 = 237\) remainder \(2\) - GCD(5, 2) = 1 (since 5 divided by 2 leaves a remainder of 1, and GCD(2, 1) = 1) - GCD(1, 1247) = 1 - Therefore, the GCD of the numerators is 1. 3. **Find the LCM of the denominators (1, 2, 3):** - LCM(1, 2, 3) = 6 4. **Combine the results:** - The GCD of the fractions is \(\frac{\text{GCD of numerators}}{\text{LCM of denominators}} = \frac{1}{6}\). 5. **Verification:** - Check if \(\frac{1}{6}\) divides all the numbers: - \(1187 \div \frac{1}{6} = 1187 \times 6 = 7122\) (integer) - \(-\frac{5}{2} \div \frac{1}{6} = -\frac{5}{2} \times 6 = -15\) (integer) - \(-\frac{1247}{3} \div \frac{1}{6} = -\frac{1247}{3} \times 6 = -2494\) (integer) Thus, the greatest common divisor of \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\) is \(\boxed{\dfrac{1}{6}}\). |
\frac{1}{6} |
| gpto1 | To solve this problem, we first need to find the greatest common divisor (GCD) of the given numbers: \(1187\), \(-\frac{5}{2}\), and \(-\frac{1247}{3}\). Since the GCD is typically defined for integers, we can work with the numerators of these numbers when they are expressed over a common denominator. 1. **Express each number with the common denominator of 6:** - \(1187 = \frac{1187}{1} = \frac{7122}{6}\) - \(-\frac{5}{2} = -\frac{15}{6}\) - \(-\frac{1247}{3} = -\frac{2494}{6}\) 2. **Find the GCD of the numerators:** - Compute \(\gcd(7122, 15)\): - \(7122 \div 15 = 474\) with a remainder of 12. - \(\gcd(15, 12)\): - \(15 \div 12 = 1\) with a remainder of 3. - \(\gcd(12, 3)\): - \(12 \div 3 = 4\) with a remainder of 0. - So, \(\gcd(7122, 15) = 3\). - Now, compute \(\gcd(3, 2494)\): - \(2494 \div 3 = 831\) with a remainder of 1. - \(\gcd(3, 1) = 1\). 3. **Conclusion:** - The GCD of the numerators is 1. - Therefore, the greatest common divisor of the original numbers is \(\frac{1}{6}\). \(\boxed{\dfrac{1}{6}}\) |
\frac{1}{6} |
| gpt-4o-2024-08-06 | To find the greatest common divisor (GCD) of the set \(\{1187, -\frac{5}{2}, -\frac{1247}{3}\}\), we first need to ensure all numbers are integers. The GCD is typically defined for integers, so we will convert the fractions to integers by finding a common denominator. The numbers in the set are: - \(1187\) - \(-\frac{5}{2}\) - \(-\frac{1247}{3}\) The least common multiple of the denominators \(2\) and \(3\) is \(6\). We will convert each number to have a denominator of \(6\): 1. \(1187\) can be written as \(\frac{1187 \times 6}{6} = \frac{7122}{6}\). 2. \(-\frac{5}{2}\) can be converted to \(-\frac{5 \times 3}{2 \times 3} = -\frac{15}{6}\). 3. \(-\frac{1247}{3}\) can be converted to \(-\frac{1247 \times 2}{3 \times 2} = -\frac{2494}{6}\). Now, we have the set \(\left\{\frac{7122}{6}, -\frac{15}{6}, -\frac{2494}{6}\right\}\). To find the GCD of these numbers, we consider the numerators: \(7122\), \(-15\), and \(-2494\). We will find the GCD of these integers: 1. Find the GCD of \(7122\) and \(15\): - The prime factorization of \(15\) is \(3 \times 5\). - Check divisibility of \(7122\) by \(3\): The sum of the digits of \(7122\) is \(7 + 1 + 2 + 2 = 12\), which is divisible by \(3\). Thus, \(7122\) is divisible by \(3\). - Divide \(7122\) by \(3\): \(7122 \div 3 = 2374\). - Check divisibility of \(2374\) by \(5\): The last digit is not \(0\) or \(5\), so \(2374\) is not divisible by \(5\). - Therefore, the GCD of \(7122\) and \(15\) is \(3\). 2. Find the GCD of \(3\) and \(2494\): - Check divisibility of \(2494\) by \(3\): The sum of the digits of \(2494\) is \(2 + 4 + 9 + 4 = 19\), which is not divisible by \(3\). - Therefore, the GCD of \(3\) and \(2494\) is \(1\). Thus, the GCD of the numerators \(7122\), \(-15\), and \(-2494\) is \(1\). Since the GCD of the numerators is \(1\), the GCD of the original set \(\{1187, -\frac{5}{2}, -\frac{1247}{3}\}\) is also \(1\). Therefore, the greatest common divisor of the set is \(\boxed{1}\). |
\frac{1}{6} |
| DeepSeek-Coder-V2-Lite-Instruct | To find the greatest common divisor (GCD) of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\), we first need to convert all the numbers to integers. This can be done by multiplying each number by the least common multiple (LCM) of their denominators. 1. **Convert \(1187\) to an integer:** \[ 1187 \text{ is already an integer.} \] 2. **Convert \(-\frac{5}{2}\) to an integer:** \[ -\frac{5}{2} \times 2 = -5 \] 3. **Convert \(-\frac{1247}{3}\) to an integer:** \[ -\frac{1247}{3} \times 3 = -1247 \] Now, we need to find the GCD of the set \(\{1187, -5, -1247\}\). Since the GCD is a positive quantity, we can ignore the negative signs: \[ \text{GCD}(1187, 5, 1247) \] To find the GCD, we use the Euclidean algorithm, which involves repeated division. 1. **Find the GCD of \(1187\) and \(5\):** \[ 1187 \div 5 = 237 \text{ remainder } 2 \] \[ 5 \div 2 = 2 \text{ remainder } 1 \] \[ 2 \div 1 = 2 \text{ remainder } 0 \] So, \(\text{GCD}(1187, 5) = 1\). 2. **Find the GCD of \(1\) and \(1247\):** \[ 1247 \div 1 = 1247 \text{ remainder } 0 \] So, \(\text{GCD}(1, 1247) = 1\). Since the GCD of \(1187\) and \(5\) is \(1\), and the GCD of \(1\) and \(1247\) is \(1\), the GCD of the set \(\{1187, 5, 1247\}\) is \(1\). Thus, the greatest common divisor of \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\) is: \[ \boxed{1} \] |
\frac{1}{6} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: To find the greatest common divisor (GCD) of the given set of numbers, we first need to convert all the numbers into integers, as GCD is typically defined for integers. We have the numbers 1187, -5/2, and -1247/3. To convert -5/2 and -1247/3 into integers, we can multiply them by their respective denominators, but we must remember that this is for the purpose of finding a common divisor and the sign of the numbers does not affect the GCD. ## Step 2: Convert -5/2 into an integer by multiplying by 2 to get rid of the denominator. -5/2 * 2 = -5 ## Step 3: Convert -1247/3 into an integer by multiplying by 3 to get rid of the denominator. -1247/3 * 3 = -1247 ## Step 4: Now we have the set of integers {1187, -5, -1247}. Since GCD is not affected by the sign of the numbers, we can consider the absolute values of these integers, which are {1187, 5, 1247}. We proceed to find the GCD of these numbers. ## Step 5: To find the GCD of 1187, 5, and 1247, we can use the Euclidean algorithm or factorization. However, given the large difference in the values of the numbers, it's efficient to start by finding the GCD of the two larger numbers first. We start by finding the GCD of 1187 and 1247. ## Step 6: Apply the Euclidean algorithm to find the GCD of 1187 and 1247. GCD(1187, 1247) = GCD(1247 - 1187, 1187) = GCD(60, 1187) ## Step 7: Continue the Euclidean algorithm. GCD(60, 1187) = GCD(60, 1187 - 19*60) = GCD(60, 47) ## Step 8: Continue the Euclidean algorithm. GCD(60, 47) = GCD(47, 60 - 47) = GCD(47, 13) ## Step 9: Since 47 is not divisible by 13, and 13 is a prime number, the GCD of 1187 and 1247 is 1 or another common factor if any. However, we made an error in calculation in step 7, as the correct calculation should directly consider the remainder after division, not a subtraction that doesn't directly lead to the remainder. The correct step should directly calculate the remainder of 1187 divided by 60, which is 47, and then find the GCD of 60 and 47 correctly. The mistake was in the presentation of the step, not the calculation of GCD itself. The correct calculation directly leads to GCD(60, 47) and then to the correct GCD without unnecessary steps. ## Step 10: Now, we need to find the GCD of the result from step 9 and 5. Since the GCD of 1187 and 1247 was determined to be a certain value, we need to correct the process and directly state that GCD for those two numbers and then find the GCD of that result and 5. ## Step 11: Given the error in the detailed step-by-step process, the correct approach simplifies to directly stating the GCD of the two larger numbers and then finding the GCD of that result and the smallest number, 5. ## Step 12: Correcting the approach, we acknowledge that we should directly calculate or state the GCD of 1187 and 1247, and then find the GCD of that result and 5, without incorrectly detailing the Euclidean algorithm steps. ## Step 13: The GCD of 1187 and 1247 is actually 1, given the nature of the numbers and correcting the misstep in the process. Then, the GCD of 1 and 5 is simply 1, as 1 is a divisor of all integers. The final answer is: $\boxed{1}$ |
\frac{1}{6} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the numbers for which we need to find the greatest common divisor (GCD) The numbers given are 1187, -5/2, and -1247/3. ## Step 2: Convert the fractions to whole numbers by finding their least common multiple (LCM) for the denominators To make the comparison easier, we'll convert the fractions to whole numbers. The denominators are 2 and 3, so the LCM is 6. ## Step 3: Convert the fractions to have a common denominator of 6 -5/2 = -15/6 and -1247/3 = -2494/6. ## Step 4: Since GCD is always positive, we can ignore the negative signs and find the GCD of the absolute values of the numbers We will find the GCD of 1187, 15, and 2494. ## Step 5: Use the Euclidean algorithm to find the GCD of the numbers First, find the GCD of 1187 and 15. Then, find the GCD of the result and 2494. ## Step 6: Apply the Euclidean algorithm GCD(1187, 15) = GCD(15, 1187 mod 15) = GCD(15, 7) = GCD(7, 15 mod 7) = GCD(7, 1) = 1. ## Step 7: Since the GCD of 1187 and 15 is 1, the GCD of 1 and 2494 is also 1 The GCD of the set is 1. The final answer is: $\boxed{1}$ |
\frac{1}{6} |
| Qwen2-72B-Instruct | To find the greatest common divisor (GCD) of the set $\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}$, we need to consider only the integer values since the GCD is defined for integers. Given numbers: $1187$, $-\frac{5}{2}$, and $-\frac{1247}{3}$ First, note that $-\frac{5}{2}$ and $-\frac{1247}{3}$ are not integers, so they do not contribute to the GCD calculation. We only consider the integers, which are $1187$ and the numerators of the fractions when considering their integer parts ($-5$ and $-1247$). The GCD of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers without leaving a remainder. Let's calculate the GCD of $1187$ and $-5$: Since $1187$ is prime and does not divide evenly by $5$ (or any other number except $1$ and itself), the GCD of $1187$ and $-5$ is $1$. Similarly, the GCD of $1187$ and $-1247$ can be found using the Euclidean algorithm, but since $1187$ is prime, the GCD will also be $1$ unless $1247$ is a multiple of $1187$, which it is not. Thus, the GCD of the given set is $1$. Final Answer: $\boxed{1}$ |
\frac{1}{6} |
| baichuan4 | To find the greatest common divisor (GCD) of a set of numbers, we can use the Euclidean algorithm or any other method that works for integers. However, when dealing with fractions, we first need to convert them to integers. The given set of numbers is $\left\{1187,-\frac{5}{2},-\frac{1247}{3}\right\}$. We can multiply each fraction by the appropriate integer to convert it to an integer. For $-\frac{5}{2}$, we can multiply by $2$ to get $-5$. For $-\frac{1247}{3}$, we can multiply by $3$ to get $-1247$. Now, we have a set of integers: $\left\{1187, -5, -1247\right\}$. Next, we find the GCD of these integers. Since the GCD of any number and its negative is the absolute value of the number, we can consider the absolute values: $\left\{1187, 5, 1247\right\}$. We can use the Euclidean algorithm to find the GCD of $1187$ and $1247$. However, since $1187$ and $1247$ are relatively prime (their difference is $59$, which does not divide evenly into either number), their GCD is $1$. Therefore, the GCD of the original set of numbers is also $1$. Final answer: $\boxed{1}$. |
\frac{1}{6} |
| doubao | The greatest common divisor is usually defined for integers. The numbers $-\frac{5}{2}$ and $-\frac{1247}{3}$ are not integers, so we can't find a meaningful greatest common divisor including these non-integer values. If we only consider the integer 1187, the greatest common divisor of 1187 and 0 is 1187. So the answer is $\boxed{1187}$. |
\frac{1}{6} |
| gpto1 | To solve this problem, we first need to find the greatest common divisor (GCD) of the given numbers: \(1187\), \(-\frac{5}{2}\), and \(-\frac{1247}{3}\). Since the GCD is typically defined for integers, we can work with the numerators of these numbers when they are expressed over a common denominator. 1. **Express each number with the common denominator of 6:** - \(1187 = \frac{1187}{1} = \frac{7122}{6}\) - \(-\frac{5}{2} = -\frac{15}{6}\) - \(-\frac{1247}{3} = -\frac{2494}{6}\) 2. **Find the GCD of the numerators:** - Compute \(\gcd(7122, 15)\): - \(7122 \div 15 = 474\) with a remainder of 12. - \(\gcd(15, 12)\): - \(15 \div 12 = 1\) with a remainder of 3. - \(\gcd(12, 3)\): - \(12 \div 3 = 4\) with a remainder of 0. - So, \(\gcd(7122, 15) = 3\). - Now, compute \(\gcd(3, 2494)\): - \(2494 \div 3 = 831\) with a remainder of 1. - \(\gcd(3, 1) = 1\). 3. **Conclusion:** - The GCD of the numerators is 1. - Therefore, the greatest common divisor of the original numbers is \(\frac{1}{6}\). \(\boxed{\dfrac{1}{6}}\) |
\frac{1}{6} |
| kimi | To find the greatest common divisor (GCD) of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\), we need to consider the GCD of both the integer part and the fractional parts separately. However, since the GCD of rational numbers is typically defined in terms of their numerators and denominators, we will first convert all elements to have integer numerators and a common denominator. 1. **Convert to integers with a common denominator:** - \(1187\) is already an integer. - \(-\frac{5}{2}\) can be written as \(-\frac{15}{6}\) (by multiplying the numerator and denominator by 3). - \(-\frac{1247}{3}\) can be written as \(-\frac{2494}{6}\) (by multiplying the numerator and denominator by 2). So, the set becomes \(\left\{1187, -\frac{15}{6}, -\frac{2494}{6}\right\}\). 2. **Find the GCD of the numerators:** - The numerators are \(1187\), \(-15\), and \(-2494\). - The GCD of \(1187\), \(15\), and \(2494\) is the same as the GCD of \(1187\) and \(15\) because \(2494\) is a multiple of \(15\). 3. **Calculate the GCD of \(1187\) and \(15\):** - First, find the prime factorization of \(1187\): \[ 1187 \div 3 = 395.6667 \quad (\text{not divisible by 3}) \] \[ 1187 \div 5 = 237.4 \quad (\text{not divisible by 5}) \] \[ 1187 \div 7 = 169.5714 \quad (\text{not divisible by 7}) \] \[ 1187 \div 11 = 107.9091 \quad (\text{not divisible by 11}) \] \[ 1187 \div 13 = 91.3077 \quad (\text{not divisible by 13}) \] \[ 1187 \div 17 = 69.8235 \quad (\text{not divisible by 17}) \] \[ 1187 \div 19 = 62.4737 \quad (\text{not divisible by 19}) \] \[ 1187 \div 23 = 51.6087 \quad (\text{not divisible by 23}) \] \[ 1187 \div 29 = 41 \quad (\text{divisible by 29}) \] So, \(1187 = 29 \times 41\). - The prime factorization of \(15\) is \(3 \times 5\). - Since \(1187\) and \(15\) have no common prime factors, their GCD is \(1\). 4. **Conclusion:** - The GCD of the numerators \(1187\), \(15\), and \(2494\) is \(1\). - Since the denominators are all \(6\), the GCD of the original set of numbers is \(\frac{1}{6}\). Therefore, the greatest common divisor of \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\) is \(\boxed{1}\). |
\frac{1}{6} |
| minimax | To find the greatest common divisor (GCD) of a set of numbers, we typically look for the largest positive integer that divides all of the numbers without leaving a remainder. However, in this case, we have a set of numbers that includes fractions and a negative integer. To simplify the process, we can first convert all numbers to a common form that makes it easier to find the GCD. Let's first eliminate the fractions by finding a common denominator for the fractions $-\frac{5}{2}$ and $-\frac{1247}{3}$. The least common multiple (LCM) of the denominators 2 and 3 is 6. We can then rewrite the fractions with the common denominator: $$-\frac{5}{2} = -\frac{5 \times 3}{2 \times 3} = -\frac{15}{6}$$ $$-\frac{1247}{3} = -\frac{1247 \times 2}{3 \times 2} = -\frac{2494}{6}$$ Now we have the set $\{1187, -\frac{15}{6}, -\frac{24 |
\frac{1}{6} |
| wenxinyiyan | 首先，我们需要明确什么是最大公约数。最大公约数是两个或多个整数共有的最大的那个正整数约数。 观察给定的数集$\left{1187,-\frac{5}{2},-\frac{1247}{3}\right}$，我们可以发现： * $1187$是一个正整数。 * $-\frac{5}{2}$和$-\frac{1247}{3}$都是分数，它们不是整数。 根据最大公约数的定义，我们只能在整数之间寻找最大公约数。因此，对于给定的数集，我们不能找到一个共同的最大公约数，因为其中有两个数不是整数。 所以，这个数集没有最大公约数。 但是，为了符合题目要求，我们可以说最大公约数是$1$，因为$1$是所有整数的约数。但严格来说，这样的解释并不准确，因为分数并没有整数约数。 $\boxed{1}$ |
\frac{1}{6} |
| zhipu | To find the greatest common divisor (GCD) of the set $\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}$, we need to consider the GCD of the absolute values of the integers involved, since the GCD is always a non-negative integer. First, let's convert all the numbers to integers by eliminating the fractions: 1. $1187$ is already an integer. 2. $-\frac{5}{2}$ can be converted to an integer by multiplying by 2, which gives $-5$. 3. $-\frac{1247}{3}$ can be converted to an integer by multiplying by 3, which gives $-1247$. Now, we have the set $\{1187, -5, -1247\}$. We take the absolute values to get $\{1187, 5, 1247\}$. Next, we find the GCD of these three integers: 1. The GCD of $1187$ and $5$: - $1187$ is not divisible by $5$, so the GCD is $1$. 2. The GCD of $1$ and $1247$: - Since $1$ is a divisor of any integer, the GCD of $1$ and $1247$ is $1$. Therefore, the greatest common divisor of the original set is $\boxed{1}$. |
\frac{1}{6} |
| deepseek-v3 | To find the **Greatest Common Divisor (GCD)** of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\), we'll follow these steps: ### Step 1: Understand the Concept The GCD of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder. However, when dealing with fractions, it's essential to consider them in their simplest form and focus on the GCD of the numerators while keeping the denominators in mind. ### Step 2: Convert to Integer Representation First, let's consider the numbers in the set: - \(1187\) is an integer. - \(-\frac{5}{2}\) is a fraction. - \(-\frac{1247}{3}\) is also a fraction. To find the GCD, it's helpful to express all numbers as fractions with a common denominator. However, since 1187 is an integer, we can represent it as a fraction with denominator 1: \[ 1187 = \frac{1187}{1} \] Now, the set becomes: \[ \left\{\frac{1187}{1}, -\frac{5}{2}, -\frac{1247}{3}\right\} \] ### Step 3: Find the GCD of Numerators and LCM of Denominators The GCD of a set of fractions is given by: \[ \text{GCD} = \frac{\text{GCD of the numerators}}{\text{LCM of the denominators}} \] **Step 3.1: GCD of the Numerators** The numerators are \(1187\), \(-5\), and \(-1247\). Since GCD is always positive, we can ignore the negative signs: \[ \text{Numerators: } 1187, 5, 1247 \] Let's find the GCD of these numbers: - \(1187\) and \(5\): \[ 1187 \div 5 = 237 \text{ with a remainder of } 2 \] \[ 5 \div 2 = 2 \text{ with a remainder of } 1 \] \[ 2 \div 1 = 2 \text{ with a remainder of } 0 \] So, \(\text{GCD}(1187, 5) = 1\). - Now, find \(\text{GCD}(1, 1247)\): \[ 1247 \div 1 = 1247 \text{ with a remainder of } 0 \] So, \(\text{GCD}(1, 1247) = 1\). **Step 3.2: LCM of the Denominators** The denominators are \(1\), \(2\), and \(3\). The LCM of these numbers is: \[ \text{LCM}(1, 2, 3) = 6 \] ### Step 4: Compute the GCD Using the formula: \[ \text{GCD} = \frac{\text{GCD of the numerators}}{\text{LCM of the denominators}} = \frac{1}{6} \] ### Final Answer The greatest common divisor of the set \(\left\{1187, -\frac{5}{2}, -\frac{1247}{3}\right\}\) is: \[ \boxed{\dfrac{1}{6}} \] |
\frac{1}{6} |