| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we follow these steps: Step 1: Divide 10236 by 244 and find the remainder. 10236 ÷ 244 = 42 with a remainder of 0. Since the remainder is 0, this indicates that 244 is a divisor of 10236. Therefore, the gcd(10236, 244) = 244. |
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| deepseekr1 | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, follow these steps: 1. **Divide 10236 by 244**: \(10236 = 244 \times 41 + 232\) Remainder = 232 Now, compute gcd(244, 232). 2. **Divide 244 by 232**: \(244 = 232 \times 1 + 12\) Remainder = 12 Compute gcd(232, 12). 3. **Divide 232 by 12**: \(232 = 12 \times 19 + 4\) Remainder = 4 Compute gcd(12, 4). 4. **Divide 12 by 4**: \(12 = 4 \times 3 + 0\) Remainder = 0 The last non-zero remainder is **4**, so the gcd is **4**. **Verification via Prime Factorization**: - \(10236 = 2^2 \times 3 \times 853\) - \(244 = 2^2 \times 61\) - Common factors: \(2^2 = 4\) **Answer**: \(\boxed{4}\) |
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| gpto1 | To find the greatest common divisor (gcd) of **10236** and **244** using the Euclidean algorithm, follow these steps: ### Step 1: Divide the larger number by the smaller number Divide **10236** by **244** and find the remainder. \[ \begin{align*} 10236 \div 244 &= 41 \ \text{with a remainder of} \ 232 \\ \text{So}, \quad 10236 &= 244 \times 41 + 232 \end{align*} \] ### Step 2: Replace the larger number with the smaller number, and the smaller number with the remainder Now, take **244** and **232** and apply the algorithm again. \[ \begin{align*} 244 \div 232 &= 1 \ \text{with a remainder of} \ 12 \\ \text{So}, \quad 244 &= 232 \times 1 + 12 \end{align*} \] ### Step 3: Repeat the process Take **232** and **12** and continue. \[ \begin{align*} 232 \div 12 &= 19 \ \text{with a remainder of} \ 4 \\ \text{So}, \quad 232 &= 12 \times 19 + 4 \end{align*} \] ### Step 4: Continue until the remainder is zero Now, take **12** and **4**. \[ \begin{align*} 12 \div 4 &= 3 \ \text{with a remainder of} \ 0 \\ \text{So}, \quad 12 &= 4 \times 3 + 0 \end{align*} \] ### Conclusion The last non-zero remainder is **4**, which is the greatest common divisor of **10236** and **244**. **Answer:** The greatest common divisor of **10236** and **244** is **4**. |
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| GPT-4o | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we follow these steps: 1. **Divide 10236 by 244 and find the remainder.** \[ 10236 \div 244 = 41 \quad \text{remainder } 172 \] So, \( 10236 = 244 \times 41 + 172 \). 2. **Replace 10236 with 244 and 244 with the remainder 172, then repeat the process.** \[ 244 \div 172 = 1 \quad \text{remainder } 72 \] So, \( 244 = 172 \times 1 + 72 \). 3. **Replace 244 with 172 and 172 with the remainder 72, then repeat the process.** \[ 172 \div 72 = 2 \quad \text{remainder } 28 \] So, \( 172 = 72 \times 2 + 28 \). 4. **Replace 172 with 72 and 72 with the remainder 28, then repeat the process.** \[ 72 \div 28 = 2 \quad \text{remainder } 16 \] So, \( 72 = 28 \times 2 + 16 \). 5. **Replace 72 with 28 and 28 with the remainder 16, then repeat the process.** \[ 28 \div 16 = 1 \quad \text{remainder } 12 \] So, \( 28 = 16 \times 1 + 12 \). 6. **Replace 28 with 16 and 16 with the remainder 12, then repeat the process.** \[ 16 \div 12 = 1 \quad \text{remainder } 4 \] So, \( 16 = 12 \times 1 + 4 \). 7. **Replace 16 with 12 and 12 with the remainder 4, then repeat the process.** \[ 12 \div 4 = 3 \quad \text{remainder } 0 \] So, \( 12 = 4 \times 3 + 0 \). Since the remainder is now 0, the gcd is the last non-zero remainder, which is 4. Thus, the gcd of 10236 and 244 is \( \boxed{4} \). |
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| llama3.1-70B-Instruct | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we follow these steps: 1. Divide the larger number by the smaller number and find the remainder. 2. Replace the larger number with the smaller number and the smaller number with the remainder. 3. Repeat steps 1 and 2 until the remainder is 0. Let's apply the Euclidean algorithm to 10236 and 244: 1. 10236 ÷ 244 = 42 with a remainder of 48. 2. 244 ÷ 48 = 5 with a remainder of 4. 3. 48 ÷ 4 = 12 with a remainder of 0. Since the remainder is now 0, the gcd of 10236 and 244 is the last non-zero remainder, which is 4. Therefore, the gcd of 10236 and 244 is 4. |
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| llama3.1-405B-Instruct | To find the greatest common divisor (GCD) of 10236 and 244 using the Euclidean algorithm, we follow these steps: 1. Divide the larger number by the smaller number and find the remainder. 2. Replace the larger number with the smaller number and the smaller number with the remainder. 3. Repeat steps 1 and 2 until the remainder is 0. Here are the steps for the given numbers: 1. 10236 ÷ 244 = 41 remainder 212 2. 244 ÷ 212 = 1 remainder 32 3. 212 ÷ 32 = 6 remainder 20 4. 32 ÷ 20 = 1 remainder 12 5. 20 ÷ 12 = 1 remainder 8 6. 12 ÷ 8 = 1 remainder 4 7. 8 ÷ 4 = 2 remainder 0 The last non-zero remainder is 4, so the GCD of 10236 and 244 is 4. |
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| DeepSeek-Coder-V2 | The Euclidean algorithm is a method for finding the greatest common divisor (gcd) of two numbers. It involves repeatedly applying the operation of replacing the larger number by its remainder when divided by the smaller number, until the remainder is zero. The last non-zero remainder is the gcd of the original two numbers. Let's apply the Euclidean algorithm to find the gcd of (10236, 244): 1. Divide 10236 by 244 and find the remainder: \[ 10236 \div 244 = 41 \quad \text{remainder} \quad 212 \] So, 10236 = 244 * 41 + 212. 2. Now, replace 10236 with 212 and repeat the process: \[ 244 \div 212 = 1 \quad \text{remainder} \quad 32 \] So, 244 = 212 * 1 + 32. 3. Replace 244 with 32 and repeat: \[ 212 \div 32 = 6 \quad \text{remainder} \quad 20 \] So, 212 = 32 * 6 + 20. 4. Replace 212 with 20 and repeat: \[ 32 \div 20 = 1 \quad \text{remainder} \quad 12 \] So, 32 = 20 * 1 + 12. 5. Replace 32 with 12 and repeat: \[ 20 \div 12 = 1 \quad \text{remainder} \quad 8 \] So, 20 = 12 * 1 + 8. 6. Replace 20 with 8 and repeat: \[ 12 \div 8 = 1 \quad \text{remainder} \quad 4 \] So, 12 = 8 * 1 + 4. 7. Replace 12 with 4 and repeat: \[ 8 \div 4 = 2 \quad \text{remainder} \quad 0 \] So, 8 = 4 * 2 + 0. Since the remainder is now 0, the last non-zero remainder is the gcd. Therefore, the gcd of (10236, 244) is 4. |
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| Qwen2-72B-Instruct | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we repeatedly apply the algorithm until we get a remainder of 0. The last non-zero remainder is the gcd. Step 1: Divide the larger number by the smaller number and find the remainder. 10236 ÷ 244 = 42 with a remainder of 0. However, since the remainder is 0 right away, this means 244 is a divisor of 10236. The gcd(10236, 244) = 244 since 244 divides 10236 exactly with no remainder. However, if the division had not resulted in a remainder of 0 immediately, we would continue the process using the divisor (244) as the new dividend and the remainder as the new divisor until we got a remainder of 0. In this case, the gcd is simply the smaller number, 244. So, gcd(10236, 244) = 244. |
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| baichuan4 | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we repeatedly apply the division algorithm to find the remainder until the remainder is zero. The last non-zero remainder will be the gcd of the two numbers. Here's the step-by-step process: 1. Divide 10236 by 244: 10236 = 244 * 42 + 4 2. Divide 244 by the remainder 4: 244 = 4 * 61 + 0 Now, the remainder is zero, so we stop. The last non-zero remainder was 4, so the gcd of 10236 and 244 is 4. |
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| doubao | Using the Euclidean algorithm: 10236 ÷ 244 = 42 remainder 0. So the greatest common divisor (GCD) of 10236 and 244 is 244. |
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| minimax | The Euclidean algorithm is a method for finding the greatest common divisor (gcd) of two numbers by repeatedly applying the division algorithm. Here's how you can use it to find the gcd of 10236 and 244: 1. Divide the larger number by the smaller number and find the remainder. 2. Replace the larger number with the smaller number and the smaller number with the remainder from step 1. 3. Repeat the process until the remainder is 0. The non-zero remainder just before this is the gcd of the two numbers. Let's apply this to the numbers 10236 and 244: Step 1: Divide 10236 by 244 and find the remainder. 10236 ÷ 244 = 41 remainder 208 Step 2: Replace 10236 with 244 and 244 with 208. Now divide 244 by 208 and find the remainder. 244 ÷ 208 = 1 remainder 36 Step 3: Replace 244 with 208 and 208 with 36. Now divide 208 by 36 and find the remainder. 208 ÷ 36 = 5 remainder 32 Step 4: Replace 208 with 36 and 36 with 32. Now divide 36 by 32 and find the remainder. 36 ÷ 32 = 1 remainder 4 Step 5: Replace 36 with 32 and 32 with 4. Now divide 32 by 4 and find the remainder. 32 ÷ 4 = 8 remainder 0 Since the remainder is now 0, the process stops here. The gcd is the last non-zero remainder, which is 4. Therefore, the greatest common divisor of 10236 and 244 is 4. |
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| kimi | The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It is based on the principle that the GCD of two numbers also divides their difference. Here's how you can apply the Euclidean algorithm to find the GCD of 10236 and 244: 1. Divide the larger number by the smaller number and find the remainder. 2. Replace the larger number with the smaller number and the smaller number with the remainder from step 1. 3. Repeat the process until the remainder is 0. The non-zero remainder just before this is the GCD. Let's apply this to 10236 and 244: 1. Divide 10236 by 244: 10236 ÷ 244 = 41 remainder 212 2. Now, take 244 and 212 and repeat the process: 244 ÷ 212 = 1 remainder 32 3. Next, take 212 and 32: 212 ÷ 32 = 6 remainder 20 4. Then, take 32 and 20: 32 ÷ 20 = 1 remainder 12 5. Now, take 20 and 12: 20 ÷ 12 = 1 remainder 8 6. Finally, take 12 and 8: 12 ÷ 8 = 1 remainder 4 7. Now, take 8 and 4: 8 ÷ 4 = 2 remainder 0 Since we've reached a remainder of 0, the last non-zero remainder is the GCD. So, the GCD of 10236 and 244 is 4. |
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| wenxinyiyan | To find the greatest common divisor (gcd) of 10236 and 244 using the Euclidean algorithm, we repeatedly subtract the smaller number from the larger number until the two numbers become equal. The resulting number is the gcd. However, for larger numbers, it is more efficient to use the division version of the Euclidean algorithm, which replaces subtraction with division and takes the remainder at each step. Here's how we apply the Euclidean algorithm to find the gcd of 10236 and 244: Step 1: Divide 10236 by 244 and note the remainder. 10236 ÷ 244 = 41 with remainder 212 (So, 10236 = 244 × 41 + 212) Step 2: Now, divide 244 by the remainder obtained in Step 1 (i.e., 212) and note the new remainder. 244 ÷ 212 = 1 with remainder 32 (So, 244 = 212 × 1 + 32) Step 3: Divide the previous remainder (212) by the new remainder (32). 212 ÷ 32 = 6 with remainder 20 (So, 212 = 32 × 6 + 20) Step 4: Divide the new remainder (32) by the latest remainder (20). 32 ÷ 20 = 1 with remainder 12 (So, 32 = 20 × 1 + 12) Step 5: Divide the latest remainder (20) by the new remainder (12). 20 ÷ 12 = 1 with remainder 8 (So, 20 = 12 × 1 + 8) Step 6: Divide the new remainder (12) by the latest remainder (8). 12 ÷ 8 = 1 with remainder 4 (So, 12 = 8 × 1 + 4) Step 7: Divide the latest remainder (8) by the new remainder (4). 8 ÷ 4 = 2 with remainder 0 (So, 8 = 4 × 2 + 0) Since the remainder is now 0, we stop the process. The gcd is the non-zero remainder from the previous step, which is 4. Therefore, the gcd of 10236 and 244 is 4. |
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| zhipu | The Euclidean algorithm is a method for finding the greatest common divisor (gcd) of two integers based on the principle that the gcd of two numbers also divides their difference. To find the gcd of 10236 and 244 using the Euclidean algorithm, we proceed as follows: 1. Divide 10236 by 244 and find the remainder. \( 10236 \div 244 = 41 \) with a remainder of \( 10236 - 41 \times 244 = 10236 - 10084 = 152 \). 2. Now we take 244 and the remainder from the previous step (152) and repeat the process. \( 244 \div 152 = 1 \) with a remainder of \( 244 - 1 \times 152 = 244 - 152 = 92 \). 3. Repeat again with 152 and 92. \( 152 \div 92 = 1 \) with a remainder of \( 152 - 1 \times 92 = 152 - 92 = 60 \). 4. Continue with 92 and 60. \( 92 \div 60 = 1 \) with a remainder of \( 92 - 1 \times 60 = 92 - 60 = 32 \). 5. Continue with 60 and 32. \( 60 \div 32 = 1 \) with a remainder of \( 60 - 1 \times 32 = 60 - 32 = 28 \). 6. Continue with 32 and 28. \( 32 \div 28 = 1 \) with a remainder of \( 32 - 1 \times 28 = 32 - 28 = 4 \). 7. Continue with 28 and 4. \( 28 \div 4 = 7 \) with a remainder of \( 28 - 7 \times 4 = 28 - 28 = 0 \). Since the remainder is now 0, the process stops, and the gcd is the last non-zero remainder, which is 4. Therefore, the gcd of 10236 and 244 is 4. |
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| gpto1 | To find the greatest common divisor (gcd) of **10236** and **244** using the Euclidean algorithm, follow these steps: ### Step 1: Divide the larger number by the smaller number Divide **10236** by **244** and find the remainder. \[ \begin{align*} 10236 \div 244 &= 41 \ \text{with a remainder of} \ 232 \\ \text{So}, \quad 10236 &= 244 \times 41 + 232 \end{align*} \] ### Step 2: Replace the larger number with the smaller number, and the smaller number with the remainder Now, take **244** and **232** and apply the algorithm again. \[ \begin{align*} 244 \div 232 &= 1 \ \text{with a remainder of} \ 12 \\ \text{So}, \quad 244 &= 232 \times 1 + 12 \end{align*} \] ### Step 3: Repeat the process Take **232** and **12** and continue. \[ \begin{align*} 232 \div 12 &= 19 \ \text{with a remainder of} \ 4 \\ \text{So}, \quad 232 &= 12 \times 19 + 4 \end{align*} \] ### Step 4: Continue until the remainder is zero Now, take **12** and **4**. \[ \begin{align*} 12 \div 4 &= 3 \ \text{with a remainder of} \ 0 \\ \text{So}, \quad 12 &= 4 \times 3 + 0 \end{align*} \] ### Conclusion The last non-zero remainder is **4**, which is the greatest common divisor of **10236** and **244**. **Answer:** The greatest common divisor of **10236** and **244** is **4**. |
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| deepseekv3 | To find the greatest common divisor (gcd) of \(10236\) and \(244\) using the Euclidean algorithm, follow these steps: --- ### Step 1: Apply the Euclidean algorithm The Euclidean algorithm is based on the principle that: \[ \gcd(a, b) = \gcd(b, a \mod b) \] where \(a \mod b\) is the remainder when \(a\) is divided by \(b\). --- ### Step 2: Perform the divisions 1. Start with \(a = 10236\) and \(b = 244\). \[ 10236 \div 244 = 41 \quad \text{with a remainder of} \quad 10236 - (244 \cdot 41) = 232 \] Now, \(\gcd(10236, 244) = \gcd(244, 232)\). 2. Next, compute \(244 \div 232\): \[ 244 \div 232 = 1 \quad \text{with a remainder of} \quad 244 - (232 \cdot 1) = 12 \] Now, \(\gcd(244, 232) = \gcd(232, 12)\). 3. Next, compute \(232 \div 12\): \[ 232 \div 12 = 19 \quad \text{with a remainder of} \quad 232 - (12 \cdot 19) = 4 \] Now, \(\gcd(232, 12) = \gcd(12, 4)\). 4. Next, compute \(12 \div 4\): \[ 12 \div 4 = 3 \quad \text{with a remainder of} \quad 12 - (4 \cdot 3) = 0 \] Now, \(\gcd(12, 4) = \gcd(4, 0)\). 5. Since the remainder is \(0\), the process stops. The gcd is the last non-zero remainder. --- ### Step 3: Result The greatest common divisor of \(10236\) and \(244\) is: \[ \gcd(10236, 244) = 4 \] --- ### Final Answer: \[ \boxed{4} \] |
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