| 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 \(\left\{-1052, -\frac{5315}{7}, 849\right\}\), we need to follow these steps: 1. **Convert all numbers to integers if necessary**: - \(-1052\) is already an integer. - \(-\frac{5315}{7}\) is not an integer, but we can multiply all numbers by 7 to clear the fraction. - \(849\) is already an integer. Multiplying each number by 7: - \(-1052 \times 7 = -7364\) - \(-\frac{5315}{7} \times 7 = -5315\) - \(849 \times 7 = 5943\) 2. **Find the GCD of the new set of integers**: - The new set is \(\{-7364, -5315, 5943\}\). 3. **Use the Euclidean algorithm to find the GCD**: - First, find the GCD of \(-7364\) and \(-5315\): \[ \text{GCD}(-7364, -5315) = \text{GCD}(7364, 5315) \] Using the Euclidean algorithm: \[ 7364 \div 5315 = 1 \quad \text{remainder} \quad 7364 - 5315 = 2049 \] \[ \text{GCD}(7364, 5315) = \text{GCD}(5315, 2049) \] \[ 5315 \div 2049 = 2 \quad \text{remainder} \quad 5315 - 2 \times 2049 = 1217 \] \[ \text{GCD}(5315, 2049) = \text{GCD}(2049, 1217) \] \[ 2049 \div 1217 = 1 \quad \text{remainder} \quad 2049 - 1217 = 832 \] \[ \text{GCD}(2049, 1217) = \text{GCD}(1217, 832) \] \[ 1217 \div 832 = 1 \quad \text{remainder} \quad 1217 - 832 = 385 \] \[ \text{GCD}(1217, 832) = \text{GCD}(832, 385) \] \[ 832 \div 385 = 2 \quad \text{remainder} \quad 832 - 2 \times 385 = 62 \] \[ \text{GCD}(832, 385) = \text{GCD}(385, 62) \] \[ 385 \div 62 = 6 \quad \text{remainder} \quad 385 - 6 \times 62 = 13 \] \[ \text{GCD}(385, 62) = \text{GCD}(62, 13) \] \[ 62 \div 13 = 4 \quad \text{remainder} \quad 62 - 4 \times 13 = 10 \] \[ \text{GCD}(62, 13) = \text{GCD}(13, 10) \] \[ 13 \div 10 = 1 \quad \text{remainder} \quad 13 - 10 = 3 \] \[ \text{GCD}(13, 10) = \text{GCD}(10, 3) \] \[ 10 \div 3 = 3 \quad \text{remainder} \quad 10 - 3 \times 3 = 1 \] \[ \text{GCD}(10, 3) = \text{GCD}(3, 1) \] \[ \text{GCD}(3, 1) = 1 \] - Now, find the GCD of the result (1) and the third number (5943): \[ \text{GCD}(1, 5943) = 1 \] Therefore, the greatest common divisor of \(\left\{-1052, -\frac{5315}{7}, 849\right\}\) is \(\boxed{1}\). |
\frac{1}{7} |
| deepseek-r1 | To find the greatest common divisor (GCD) of the set \(\left\{-1052, -\frac{5315}{7}, 849\right\}\), we need to consider the absolute values and handle the fraction appropriately. First, we convert all numbers to fractions: - \(-1052\) can be written as \(-\frac{1052}{1}\) - \(-\frac{5315}{7}\) is already a fraction - \(849\) can be written as \(\frac{849}{1}\) The GCD of these fractions is given by the GCD of the numerators divided by the least common multiple (LCM) of the denominators. The numerators are \(1052\), \(5315\), and \(849\), and the denominators are \(1\), \(7\), and \(1\). 1. Compute the GCD of the numerators \(1052\), \(5315\), and \(849\): - Using the Euclidean algorithm, we find that \(\gcd(1052, 5315) = 1\). - Then, \(\gcd(1, 849) = 1\). 2. Compute the LCM of the denominators \(1\), \(7\), and \(1\): - The LCM is \(7\). Thus, the GCD of the original set is \(\frac{\gcd(1052, 5315, 849)}{\text{LCM}(1, 7, 1)} = \frac{1}{7}\). Therefore, the greatest common divisor of \(\left\{-1052, -\frac{5315}{7}, 849\right\}\) is \(\boxed{\dfrac{1}{7}}\). |
\frac{1}{7} |
| gpto1 | To find the greatest common divisor (gcd) of the set \(\{-1052, -\dfrac{5315}{7}, 849\}\), let's first eliminate the fraction to work with integers, as the gcd is defined for integers. First, multiply each number by \(7\) to eliminate the denominators: \[ -1052 \times 7 = -7364 \\ -\dfrac{5315}{7} \times 7 = -5315 \\ 849 \times 7 = 5943 \] So the set becomes \(\{-7364, -5315, 5943\}\). Now, we will find the gcd of these three integers. ### Finding \(\gcd(7364, 5315)\): We use the Euclidean algorithm, considering the absolute values. 1. \(7364 = 5315 \times 1 + 2049\) 2. \(5315 = 2049 \times 2 + 1217\) 3. \(2049 = 1217 \times 1 + 832\) 4. \(1217 = 832 \times 1 + 385\) 5. \(832 = 385 \times 2 + 62\) 6. \(385 = 62 \times 6 + 13\) 7. \(62 = 13 \times 4 + 10\) 8. \(13 = 10 \times 1 + 3\) 9. \(10 = 3 \times 3 + 1\) 10. \(3 = 1 \times 3 + 0\) Since the last non-zero remainder is \(1\), \(\gcd(7364, 5315) = 1\). ### Finding \(\gcd(7364, 5943)\): 1. \(7364 = 5943 \times 1 + 1421\) 2. \(5943 = 1421 \times 4 + 259\) 3. \(1421 = 259 \times 5 + 126\) 4. \(259 = 126 \times 2 + 7\) 5. \(126 = 7 \times 18 + 0\) The last non-zero remainder is \(7\), so \(\gcd(7364, 5943) = 7\). Now, to find the gcd of all three numbers, we take the gcd of the gcds we found: \(\gcd\big(\gcd(7364, 5315), \gcd(7364, 5943)\big) = \gcd(1, 7) = 1\) But since one of the gcds is \(1\), and the other is \(7\), and the gcd of \(1\) and \(7\) is \(1\), the overall gcd is \(1\). However, since \(7\) is a common factor in \(7364\) and \(5943\) but not in \(5315\), and given that \(5315 = 7 \times 759 + 2\) (not divisible by \(7\)), but \(5943\) and \(7364\) are multiples of \(7\), the greatest common divisor of the original set is \(1\). But considering the problem involves the fraction \(-\dfrac{5315}{7}\), it's clear that \(7\) plays a key role. So, the greatest common divisor of the original set is \(\boxed{7}\). \(\boxed{7}\) |
\frac{1}{7} |
| gpt-4o-2024-08-06 | To find the greatest common divisor (GCD) of the set \(\{-1052, -\frac{5315}{7}, 849\}\), we first simplify the numbers and then find their GCD. First, simplify \(-\frac{5315}{7}\): \[ -\frac{5315}{7} = -759 \] Thus, the set becomes \(\{-1052, -759, 849\}\). The GCD is unaffected by the signs, so we consider the absolute values: \(\{1052, 759, 849\}\). We will find the GCD of these numbers using the Euclidean algorithm. **Step 1: Find \(\gcd(1052, 759)\):** Perform the division: \[ 1052 \div 759 \approx 1.385 \] The quotient is 1, so: \[ 1052 = 759 \times 1 + 293 \] Now, find \(\gcd(759, 293)\). **Step 2: Find \(\gcd(759, 293)\):** Perform the division: \[ 759 \div 293 \approx 2.589 \] The quotient is 2, so: \[ 759 = 293 \times 2 + 173 \] Now, find \(\gcd(293, 173)\). **Step 3: Find \(\gcd(293, 173)\):** Perform the division: \[ 293 \div 173 \approx 1.693 \] The quotient is 1, so: \[ 293 = 173 \times 1 + 120 \] Now, find \(\gcd(173, 120)\). **Step 4: Find \(\gcd(173, 120)\):** Perform the division: \[ 173 \div 120 \approx 1.441 \] The quotient is 1, so: \[ 173 = 120 \times 1 + 53 \] Now, find \(\gcd(120, 53)\). **Step 5: Find \(\gcd(120, 53)\):** Perform the division: \[ 120 \div 53 \approx 2.264 \] The quotient is 2, so: \[ 120 = 53 \times 2 + 14 \] Now, find \(\gcd(53, 14)\). **Step 6: Find \(\gcd(53, 14)\):** Perform the division: \[ 53 \div 14 \approx 3.785 \] The quotient is 3, so: \[ 53 = 14 \times 3 + 11 \] Now, find \(\gcd(14, 11)\). **Step 7: Find \(\gcd(14, 11)\):** Perform the division: \[ 14 \div 11 \approx 1.273 \] The quotient is 1, so: \[ 14 = 11 \times 1 + 3 \] Now, find \(\gcd(11, 3)\). **Step 8: Find \(\gcd(11, 3)\):** Perform the division: \[ 11 \div 3 \approx 3.667 \] The quotient is 3, so: \[ 11 = 3 \times 3 + 2 \] Now, find \(\gcd(3, 2)\). **Step 9: Find \(\gcd(3, 2)\):** Perform the division: \[ 3 \div 2 \approx 1.5 \] The quotient is 1, so: \[ 3 = 2 \times 1 + 1 \] Now, find \(\gcd(2, 1)\). **Step 10: Find \(\gcd(2, 1)\):** Perform the division: \[ 2 \div 1 = 2 \] The remainder is 0, so: \[ 2 = 1 \times 2 + 0 \] Thus, \(\gcd(2, 1) = 1\). Therefore, \(\gcd(1052, 759) = 1\). Now, we need to find \(\gcd(1, 849)\). Since the GCD of any number and 1 is 1, we have: \[ \gcd(1, 849) = 1 \] Thus, the greatest common divisor of \(\{-1052, -\frac{5315}{7}, 849\}\) is \(\boxed{1}\). |
\frac{1}{7} |
| DeepSeek-Coder-V2-Lite-Instruct | To find the greatest common divisor (GCD) of the numbers \(\left\{-1052, -\frac{5315}{7}, 849\right\}\), we first need to convert all the numbers to a common form. Let's convert each number to its absolute value and simplify where possible. 1. **Convert \(-1052\) to its absolute value:** \[ | -1052 | = 1052 \] 2. **Convert \(-\frac{5315}{7}\) to its absolute value:** \[ \left| -\frac{5315}{7} \right| = \frac{5315}{7} \] 3. **The number \(849\) is already positive:** \[ | 849 | = 849 \] Now, we need to find the GCD of \(1052\), \(\frac{5315}{7}\), and \(849\). First, let's find the prime factorizations of these numbers: - **Prime factorization of \(1052\):** \[ 1052 = 2^2 \times 263 \] - **Prime factorization of \(849\):** \[ 849 = 3 \times 283 \] - **Prime factorization of \(\frac{5315}{7}\):** \[ 5315 = 5 \times 1063 \] Since \(7\) is a prime number, it remains in the denominator. Next, we identify the common prime factors among the numbers. The only common prime factor is \(1\) (since \(263\), \(283\), and \(1063\) are all distinct primes). Therefore, the greatest common divisor of \(1052\), \(\frac{5315}{7}\), and \(849\) is \(1\). Thus, the final answer is: \[ \boxed{1} \] |
\frac{1}{7} |
| 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 ensure that all the numbers are integers, as the GCD is defined for integers. ## Step 2: The given set is $\left\{-1052,-\frac{5315}{7},849\right\}$. We notice that $-\frac{5315}{7}$ is not an integer, so we need to convert it into an integer to proceed with finding the GCD. ## Step 3: To make $-\frac{5315}{7}$ an integer, we can multiply it by $7$, resulting in $-5315$. However, we must also multiply the other numbers in the set by $7$ to maintain equivalence in terms of finding the GCD. ## Step 4: Multiplying $-1052$ by $7$ gives us $-7364$, and multiplying $849$ by $7$ gives us $5943$. ## Step 5: Now, we have the set $\left\{-7364,-5315,5943\right\}$, and we can proceed to find the GCD of these integers. ## Step 6: To find the GCD, we can use the Euclidean algorithm, which involves finding the greatest common divisor of two numbers and then using that result to find the GCD of the third number with the previously found GCD. ## Step 7: We start by finding the GCD of $-7364$ and $-5315$. Using the Euclidean algorithm, we can find that the GCD of these two numbers is $-133$. ## Step 8: Next, we find the GCD of $-133$ and $5943$. Again, using the Euclidean algorithm, we find that the GCD of these two numbers is $-7$. ## Step 9: Since we multiplied the original numbers by $7$ to make $-\frac{5315}{7}$ an integer, we need to account for this by dividing the GCD by $7$. ## Step 10: Therefore, the greatest common divisor of the original set of numbers is $\frac{-7}{7} = -1$. However, since the GCD is defined as a positive number, we take the absolute value, resulting in a GCD of $1$. The final answer is: $\boxed{1}$ |
\frac{1}{7} |
| Meta-Llama-3.1-70B-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 to integers if they are not already. We have the numbers -1052, -5315/7, and 849. The first and the third numbers are already integers, but the second one is a fraction. ## Step 2: Convert the fraction -5315/7 to an integer by multiplying it by 7. Multiplying -5315/7 by 7 gives us -5315, which is an integer. ## Step 3: Now that we have all the numbers as integers, we can proceed to find the GCD. However, since the numbers are negative, we need to consider their absolute values to find the GCD. ## Step 4: Take the absolute values of the numbers. The absolute values of -1052, -5315, and 849 are 1052, 5315, and 849, respectively. ## Step 5: Find the prime factorization of each number to determine the GCD. We need to find the prime factors of 1052, 5315, and 849. ## Step 6: Prime factorization of 1052 is 2^2 * 263, 5315 is 5 * 1063, and 849 is 3 * 283. Now we can see the prime factors of each number. ## Step 7: Identify the common prime factors among the numbers. There are no common prime factors among 1052, 5315, and 849. ## Step 8: Since there are no common prime factors, the GCD is 1. When there are no common prime factors, the GCD is always 1. The final answer is: $\boxed{1}$ |
\frac{1}{7} |
| Qwen2-72B-Instruct | To find the greatest common divisor (GCD) of the given numbers, we first need to convert all numbers into integers. The second number is already in fractional form, so we multiply it by 7 to get an integer. Given numbers: \(-1052\), \(-\frac{5315}{7}\), \(849\) Multiplying the second number by 7 gives us: \(-5315\) So, the numbers we will be working with are: \(-1052\), \(-5315\), \(849\) The GCD of any set of numbers is unaffected by their signs, so we can ignore the negative signs for this calculation. Now, let's find the GCD of \(1052\), \(5315\), and \(849\). First, we find the GCD of the first two numbers using the Euclidean algorithm: \[ \text{GCD}(1052, 5315) \] \[ = \text{GCD}(1052, 5315 \mod 1052) \] \[ = \text{GCD}(1052, 161) \] \[ = \text{GCD}(161, 1052 \mod 161) \] \[ = \text{GCD}(161, 109) \] \[ = \text{GCD}(109, 161 \mod 109) \] \[ = \text{GCD}(109, 52) \] \[ = \text{GCD}(52, 109 \mod 52) \] \[ = \text{GCD}(52, 5) \] \[ = \text{GCD}(5, 52 \mod 5) \] \[ = \text{GCD}(5, 2) \] \[ = \text{GCD}(2, 5 \mod 2) \] \[ = \text{GCD}(2, 1) \] \[ = 1 \] Since the GCD of the first two numbers is \(1\), and \(1\) is also a divisor of the third number (\(849\)), the GCD of all three numbers is \(1\). Therefore, the greatest common divisor of \(\left\{-1052,-\frac{5315}{7},849\right\}\) is \(\boxed{1}\). |
\frac{1}{7} |
| baichuan4 | To find the greatest common divisor (GCD) of a set of numbers, we can use the Euclidean algorithm or any other method that helps us identify the largest integer that divides all the numbers in the set without leaving a remainder. The given set of numbers is $\left\{-1052,-\frac{5315}{7},849\right\}$. First, let's simplify the second number by dividing it by 7: $$-\frac{5315}{7} = -759.$$ Now, we have the set $\left\{-1052,-759,849\right\}$. Next, we observe that the first two numbers are negative, but the GCD of a set of numbers is not affected by the signs of the numbers. We can ignore the signs and work with the absolute values. So, we consider the set $\left\{1052,759,849\right\}$. We can use the Euclidean algorithm to find the GCD of these numbers. However, we can also notice that all three numbers are divisible by 3, so we can start by dividing each number by 3 to simplify the problem: $$1052 \div 3 = 350 + \frac{2}{3},$$ $$759 \div 3 = 253,$$ $$849 \div 3 = 283.$$ Since we are looking for an integer GCD, we only consider the integer parts after division, we, we now need to find the GCD of the set $\left\{350,253,283\right\}$. We can continue using the Euclidean algorithm on these numbers. However, we can also look for common factors. Observing the numbers, we can see that they are not divisible by large primes, so the GCD is likely to be a small number. We can try dividing the numbers by small primes like 2, 5, 7, 11, etc. Upon inspection, we find that none of the numbers are divisible by 2 or 5. However, they are all divisible by 7: $$350 \div 7 = 50,$$ $$253 \div 7 = 36,$$ $$283 \div 7 = 40 + \frac{3}{7}.$$ Again, we only consider the integer parts, so now we need to find the GCD of the set $\left\{50,36,40\right\}$. These numbers are relatively small, so we can just factor them directly: $$50 = 2 \times 5^2,$$ $$36 = 2^2 \times 3^2,$$ $$40 = 2^3 \times 5.$$ The common factors are 2 and 5, so the GCD is $2 \times 5 = 10$. Therefore, the GCD of the original set $\left\{-1052,-759,849\right\}$ is $3 \times 7 \times 10 = 210$. The final answer is: $$\boxed{210}$$ |
\frac{1}{7} |
| doubao | First convert $-\frac{5315}{7}$ to an integer: $-\frac{5315}{7}=-765$. Now find the greatest common divisor of $-1052$, $-765$, and $849$. Using the Euclidean algorithm: $1052=765\times1+287$ $765=287\times2+191$ $287=191\times1+96$ $191=96\times1+95$ $96=95\times1+1$ $95=1\times95+0$. So the greatest common divisor of $-1052$ and $-765$ is 1. Now find the greatest common divisor of 1 and 849, which is 1. So the greatest common divisor of $\{-1052,-\frac{5315}{7},849\}$ is $\\boxed{1}$. |
\frac{1}{7} |
| gpto1 | To find the greatest common divisor (gcd) of the set \(\{-1052, -\dfrac{5315}{7}, 849\}\), let's first eliminate the fraction to work with integers, as the gcd is defined for integers. First, multiply each number by \(7\) to eliminate the denominators: \[ -1052 \times 7 = -7364 \\ -\dfrac{5315}{7} \times 7 = -5315 \\ 849 \times 7 = 5943 \] So the set becomes \(\{-7364, -5315, 5943\}\). Now, we will find the gcd of these three integers. ### Finding \(\gcd(7364, 5315)\): We use the Euclidean algorithm, considering the absolute values. 1. \(7364 = 5315 \times 1 + 2049\) 2. \(5315 = 2049 \times 2 + 1217\) 3. \(2049 = 1217 \times 1 + 832\) 4. \(1217 = 832 \times 1 + 385\) 5. \(832 = 385 \times 2 + 62\) 6. \(385 = 62 \times 6 + 13\) 7. \(62 = 13 \times 4 + 10\) 8. \(13 = 10 \times 1 + 3\) 9. \(10 = 3 \times 3 + 1\) 10. \(3 = 1 \times 3 + 0\) Since the last non-zero remainder is \(1\), \(\gcd(7364, 5315) = 1\). ### Finding \(\gcd(7364, 5943)\): 1. \(7364 = 5943 \times 1 + 1421\) 2. \(5943 = 1421 \times 4 + 259\) 3. \(1421 = 259 \times 5 + 126\) 4. \(259 = 126 \times 2 + 7\) 5. \(126 = 7 \times 18 + 0\) The last non-zero remainder is \(7\), so \(\gcd(7364, 5943) = 7\). Now, to find the gcd of all three numbers, we take the gcd of the gcds we found: \(\gcd\big(\gcd(7364, 5315), \gcd(7364, 5943)\big) = \gcd(1, 7) = 1\) But since one of the gcds is \(1\), and the other is \(7\), and the gcd of \(1\) and \(7\) is \(1\), the overall gcd is \(1\). However, since \(7\) is a common factor in \(7364\) and \(5943\) but not in \(5315\), and given that \(5315 = 7 \times 759 + 2\) (not divisible by \(7\)), but \(5943\) and \(7364\) are multiples of \(7\), the greatest common divisor of the original set is \(1\). But considering the problem involves the fraction \(-\dfrac{5315}{7}\), it's clear that \(7\) plays a key role. So, the greatest common divisor of the original set is \(\boxed{7}\). \(\boxed{7}\) |
\frac{1}{7} |
| kimi | To find the greatest common divisor (GCD) of the numbers \(-1052\), \(-\frac{5315}{7}\), and \(849\), we will follow these steps: 1. **Convert all numbers to positive integers:** - \(-1052\) is already an integer. - \(-\frac{5315}{7}\) can be simplified to \(-761\) because \(\frac{5315}{7} = 759.2857\), which is not an integer. However, we need to consider the absolute value for GCD calculation, so we use \(761\). - \(849\) is already a positive integer. So, we need to find the GCD of \(1052\), \(761\), and \(849\). 2. **Find the prime factorization of each number:** - **Prime factorization of \(1052\):** \[ 1052 \div 2 = 526 \\ 526 \div 2 = 263 \\ 263 \text{ is a prime number} \] So, \(1052 = 2^2 \times 263\). - **Prime factorization of \(761\):** \(761\) is a prime number. - **Prime factorization of \(849\):** \[ 849 \div 3 = 283 \\ 283 \text{ is a prime number} \] So, \(849 = 3 \times 283\). 3. **Identify the common prime factors:** - The prime factorization of \(1052\) is \(2^2 \times 263\). - The prime factorization of \(761\) is \(761\). - The prime factorization of \(849\) is \(3 \times 283\). There are no common prime factors among \(1052\), \(761\), and \(849\). 4. **Determine the GCD:** Since there are no common prime factors, the greatest common divisor is \(1\). Therefore, the greatest common divisor of \(\left\{-1052, -\frac{5315}{7}, 849\right\}\) is \(\boxed{1}\). |
\frac{1}{7} |
| minimax | To find the greatest common divisor (GCD) of the three numbers, we first need to express them in a form that makes it easier to work with. The numbers are: -1052, -5315/7, and 849. First, let's simplify the second number by multiplying both the numerator and the denominator by 7 to get rid of the fraction: -5315/7 * 7/7 = -5315. Now we have: -1052, -5315, and 849. Next, we need to find the GCD of these three integers. Since the GCD is a non-negative number, we can ignore the negative signs and just consider the absolute values of the numbers: 1052, 5315, and 849. We can use the Euclidean algorithm to find the GCD of two numbers at a time and then use the result to find the GCD with the third number. Let's start by finding the GCD of 1052 and 5315: 5315 = 1052 * 5 + 105 10 |
\frac{1}{7} |
| wenxinyiyan | 首先，我们需要将这三个数转换为整数，以便找到它们的最大公约数。 对于第一个数$-1052$，它已经是整数了。 对于第二个数$-\frac{5315}{7}$，我们可以将其乘以7并取整数部分，得到： $-\frac{5315}{7} \times 7 = -5315$ 对于第三个数$849$，它已经是整数了。 现在我们有三个整数：$-1052$，$-5315$和$849$。 为了找到这三个数的最大公约数，我们可以使用欧几里得算法。但是，由于这些数是负数或正数，我们首先需要找到它们的绝对值，然后计算这些绝对值的最大公约数。 这三个数的绝对值是：$1052$，$5315$和$849$。 现在我们可以使用欧几里得算法来找到这三个数的最大公约数： $\gcd(1052, 5315, 849) = \gcd(\gcd(1052, 5315), 849)$ 首先计算$\gcd(1052, 5315)$： $5315 = 4 \times 1052 + 1107$ $1052 = 1 \times 1107 + 941$ $1107 = 1 \times 941 + 166$ $941 = 5 \times 166 + 111$ $166 = 1 \times 111 + 55$ $111 = 2 \times 55 + 1$ $55 = 55 \times 1 + 0$ 所以，$\gcd(1052, 5315) = 1$。 接下来，我们计算$\gcd(1, 849)$： $849 = 849 \times 1 + 0$ 所以，$\gcd(1, 849) = 1$。 因此，这三个数的最大公约数是$1$。 所以，答案是$\boxed{1}$。 |
\frac{1}{7} |
| zhipu | First, we convert all numbers to integers to simplify the process of finding the greatest common divisor (GCD). The set given is $\left\{-1052, -\frac{5315}{7}, 849\right\}$. 1. Convert $-\frac{5315}{7}$ to an integer: \[ -\frac{5315}{7} = -759.2857 \approx -759 \text{ (rounding to the nearest integer)} \] However, to maintain exact values, we multiply by 7 to clear the fraction: \[ -\frac{5315}{7} \times 7 = -5315 \] So, our set of integers is $\{-1052, -5315, 849\}$. 2. Find the GCD of these integers: - Start with any two numbers, say $-1052$ and $-5315$. - Use the Euclidean algorithm: \[ \gcd(-1052, -5315) = \gcd(1052, 5315) \] \[ 5315 \div 1052 \approx 5 \quad (\text{remainder} \quad 5315 - 5 \times 1052 = 5315 - 5260 = 55) \] \[ \gcd(1052, 55) \] \[ 1052 \div 55 \approx 19 \quad (\text{remainder} \quad 1052 - 19 \times 55 = 1052 - 1045 = 7) \] \[ \gcd(55, 7) \] \[ 55 \div 7 \approx 7 \quad (\text{remainder} \quad 55 - 7 \times 7 = 55 - 49 = 6) \] \[ \gcd(7, 6) \] \[ 7 \div 6 \approx 1 \quad (\text{remainder} \quad 7 - 6 = 1) \] \[ \gcd(6, 1) = 1 \] - Now, check with the third number, 849: \[ \gcd(1, 849) = 1 \] Thus, the greatest common divisor of the set $\{-1052, -5315, 849\}$ is $\boxed{1}$. |
\frac{1}{7} |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-22d248c7-dfb3-93b6-b17f-6167dc45e7db', 'request_id': '22d248c7-dfb3-93b6-b17f-6167dc45e7db'} | \frac{1}{7} |