| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | C. 900 |
C |
| deepseekr1 | The question asks for the least perfect square with three different prime factors. A perfect square requires all prime exponents in its factorization to be even. The smallest three primes are 2, 3, and 5. To form a perfect square, each prime must have an exponent of at least 2. The minimal product is \((2 \times 3 \times 5)^2 = 30^2 = 900\), which factors as \(2^2 \times 3^2 \times 5^2\). - **A (300)**: Prime factors \(2^2 \times 3 \times 5^2\). Exponent of 3 is odd, so not a perfect square. - **B (9)**: Only one prime factor (3), insufficient. - **D (666)**: Prime factors \(2 \times 3^2 \times 37\). Exponents of 2 and 37 are odd, not a perfect square. - **C (900)**: Correctly factors as \(2^2 \times 3^2 \times 5^2\), all exponents even, and uses the smallest three primes. **Answer: C** |
C |
| gpto1 | C | C |
| GPT-4o | C. 900 | C |
| llama3.1-70B-Instruct | To find the least perfect square with 3 different prime factors, we need to consider the smallest prime numbers and their combinations. The smallest prime numbers are 2, 3, and 5. To get a perfect square with these prime factors, we need to multiply them in such a way that each prime factor appears an even number of times. The smallest combination would be 2^2 * 3 * 5 = 60. However, 60 is not a perfect square. The next combination would be 2^2 * 3^2 * 5 = 180. However, 180 is not a perfect square either. The next combination would be 2^2 * 3^2 * 5^2 = 900. This is a perfect square, as 900 = 30^2. Therefore, the least perfect square with 3 different prime factors is 900. The correct answer is C. |
C |
| llama3.1-405B-Instruct | C. 900 |
C |
| DeepSeek-Coder-V2 | C. 900 | C |
| Qwen2-72B-Instruct | C. 900 | C |
| baichuan4 | B | C |
| doubao | C. 900. Prime factorization of 900 = 2×2×3×3×5×5, which has 3 different prime factors (2, 3, 5) and it is a perfect square. |
C |
| minimax | C. 900 To find the least perfect square with 3 different prime factors, we need to consider the smallest primes and ensure that each prime factor appears an even number of times (since a perfect square has all prime factors raised to an even power). The smallest primes are 2, 3, and 5. The least perfect square that uses these primes would be: (2^2) * (3^2) * (5^2) = 4 * 9 * 25 = 900 So, the correct answer is C. 900. |
C |
| kimi | To find the least perfect square with 3 different prime factors, we need to find the smallest product of three different prime numbers, and then square that product. The smallest prime numbers are 2, 3, and 5. Multiplying these together gives us: 2 * 3 * 5 = 30 Now, we square this product to find the perfect square: 30^2 = 900 So the least perfect square with 3 different prime factors is 900. Answer: C. 900 |
C |
| wenxinyiyan | C Explanation: A perfect square is a number that is the square of an integer. For example, 4, 9, 16, 25, etc. are perfect squares. To find the least perfect square with 3 different prime factors, we need to consider the squares of numbers that are products of 3 different primes. Let's analyze each option: A. 300 = 2² × 3 × 5² This is not a perfect square because it has an odd number of factors of 3. B. 9 = 3² This is a perfect square, but it only has one prime factor (3). C. 900 = 2² × 3² × 5² This is a perfect square because all the prime factors (2, 3, and 5) appear an even number of times. It has three different prime factors, so it satisfies the condition. D. 666 = 2 × 3² × 37 This is not a perfect square because it has an odd number of factors of 2 and 37. Therefore, the least perfect square with 3 different prime factors is 900. |
C |
| zhipu | To determine the least perfect square with 3 different prime factors, we need to understand the concept of perfect squares and prime factorization. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it is \(4^2\). To have 3 different prime factors, the number must be of the form \( p^2 \times q^2 \times r^2 \) where \( p, q, \) and \( r \) are distinct prime numbers. Let's analyze the given options: A. 300 - Prime factorization of 300: \( 300 = 2^2 \times 3 \times 5^2 \) - This has 3 different prime factors (2, 3, and 5), but it is not a perfect square because the powers of the primes are not all even. B. 9 - Prime factorization of 9: \( 9 = 3^2 \) - This is a perfect square, but it has only one prime factor (3). C. 900 - Prime factorization of 900: \( 900 = 2^2 \times 3^2 \times 5^2 \) - This has 3 different prime factors (2, 3, and 5), and all the powers are even, making it a perfect square. D. 666 - Prime factorization of 666: \( 666 = 2 \times 3^2 \times 37 \) - This has 3 different prime factors (2, 3, and 37), but it is not a perfect square because the powers of the primes are not all even. Based on the analysis, the least perfect square with 3 different prime factors is 900. Answer: C |
C |
| gpto1 | C | C |
| deepseekv3 | server error: Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-c3935d1a-34fd-939e-b2b1-0d93be25f04e', 'request_id': 'c3935d1a-34fd-939e-b2b1-0d93be25f04e'} | C |