| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | A. The best answer is A. |
B |
| deepseekr1 | To determine the remainder when the 100th Fibonacci term is divided by 4, we observe the periodic pattern (Pisano period) of Fibonacci numbers modulo 4. Listing the residues: - Term 1: 1 - Term 2: 1 - Term 3: (1+1) = 2 - Term 4: (1+2) = 3 - Term 5: (2+3) = 5 ≡ 1 mod4 - Term 6: (3+1) = 4 ≡ 0 mod4 - Term 7: (1+0) = 1 - Term 8: (0+1) = 1 - Term 9: (1+1) = 2 - Term 10: (1+2) = 3 - Term 11: (2+3) = 5 ≡ 1 mod4 - Term 12: (3+1) = 4 ≡ 0 mod4 The sequence repeats every 6 terms: **1, 1, 2, 3, 1, 0**. Thus, the Pisano period modulo 4 is 6. To find \( F_{100} \mod 4 \), compute \( 100 \mod 6 \), which gives a remainder of 4. This corresponds to the 4th term in the cycle, which is **3**. Answer: **B** |
B |
| gpto1 | Answer: B | B |
| GPT-4o | To find the remainder when the 100th term of the Fibonacci sequence is divided by 4, we first need to analyze the sequence modulo 4. Let's look at the Fibonacci sequence modulo 4: \[ \begin{align*} F_1 &\equiv 1 \pmod{4}, \\ F_2 &\equiv 1 \pmod{4}, \\ F_3 &\equiv (F_1 + F_2) \equiv 1 + 1 \equiv 2 \pmod{4}, \\ F_4 &\equiv (F_2 + F_3) \equiv 1 + 2 \equiv 3 \pmod{4}, \\ F_5 &\equiv (F_3 + F_4) \equiv 2 + 3 \equiv 5 \equiv 1 \pmod{4}, \\ F_6 &\equiv (F_4 + F_5) \equiv 3 + 1 \equiv 4 \equiv 0 \pmod{4}, \\ F_7 &\equiv (F_5 + F_6) \equiv 1 + 0 \equiv 1 \pmod{4}, \\ F_8 &\equiv (F_6 + F_7) \equiv 0 + 1 \equiv 1 \pmod{4}, \\ F_9 &\equiv (F_7 + F_8) \equiv 1 + 1 \equiv 2 \pmod{4}, \\ F_{10} &\equiv (F_8 + F_9) \equiv 1 + 2 \equiv 3 \pmod{4}. \end{align*} \] We observe that the sequence modulo 4 starts repeating with a period of 6: \[ 1, 1, 2, 3, 1, 0. \] To find the remainder of the 100th term, we find the position of the 100th term within one cycle of 6 terms: \[ 100 \mod 6 = 4. \] Thus, the 100th term corresponds to the 4th term in the repeating sequence: \[ F_4 \equiv 3 \pmod{4}. \] Therefore, the remainder when the 100th term of the Fibonacci sequence is divided by 4 is \(3\). Answer: B. 3 |
B |
| llama3.1-70B-Instruct | A First note that the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, $\ldots$ . Now, we can see that the sequence of the remainders of the terms of the Fibonacci sequence when divided by 4 is 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, $\ldots$ . This sequence has a period of 12. Since $100=12 \cdot 8+4$ , the remainder when the $100^{\text {th }}$ term of the sequence is divided by 4 is the same as the remainder when the $4^{\text {th }}$ term of the sequence is divided by 4, which is 3. Final Answer: The final answer is A. I hope it is correct. |
B |
| llama3.1-405B-Instruct | The first few terms of the Fibonacci sequence, modulo 4, are 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, $\ldots$ The sequence repeats itself every 12 terms. Since $100 \equiv 4 \pmod{12}$, the $100^{\mathrm{th}}$ term of the sequence is equivalent to the $4^{\mathrm{th}}$ term of the sequence modulo 4. Therefore, the $100^{\mathrm{th}}$ term of the sequence is equivalent to 3 modulo 4, so the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 4 is $\boxed{3}$. The best answer is B. |
B |
| DeepSeek-Coder-V2 | To determine the remainder when the 100th term of the Fibonacci sequence is divided by 4, we can look for a pattern in the remainders of the Fibonacci sequence modulo 4. Let's calculate the first few terms of the Fibonacci sequence modulo 4: \[ \begin{align*} F_1 & = 1 \quad \text{(mod 4)} \\ F_2 & = 1 \quad \text{(mod 4)} \\ F_3 & = (F_1 + F_2) = (1 + 1) = 2 \quad \text{(mod 4)} \\ F_4 & = (F_2 + F_3) = (1 + 2) = 3 \quad \text{(mod 4)} \\ F_5 & = (F_3 + F_4) = (2 + 3) = 5 \equiv 1 \quad \text{(mod 4)} \\ F_6 & = (F_4 + F_5) = (3 + 1) = 4 \equiv 0 \quad \text{(mod 4)} \\ F_7 & = (F_5 + F_6) = (1 + 0) = 1 \quad \text{(mod 4)} \\ F_8 & = (F_6 + F_7) = (0 + 1) = 1 \quad \text{(mod 4)} \\ F_9 & = (F_7 + F_8) = (1 + 1) = 2 \quad \text{(mod 4)} \\ F_{10} & = (F_8 + F_9) = (1 + 2) = 3 \quad \text{(mod 4)} \\ F_{11} & = (F_9 + F_{10}) = (2 + 3) = 5 \equiv 1 \quad \text{(mod 4)} \\ F_{12} & = (F_{10} + F_{11}) = (3 + 1) = 4 \equiv 0 \quad \text{(mod 4)} \\ \end{align*} \] We observe that the sequence of remainders modulo 4 repeats every 6 terms: \(1, 1, 2, 3, 1, 0\). To find the remainder when the 100th term is divided by 4, we need to determine the position of the 100th term within this repeating cycle. We do this by finding the remainder of 100 divided by 6: \[ 100 \div 6 = 16 \text{ remainder } 4 \] This means the 100th term corresponds to the 4th term in the repeating cycle \(1, 1, 2, 3, 1, 0\). Therefore, the remainder when the 100th term of the Fibonacci sequence is divided by 4 is: \[ \boxed{B} \] |
B |
| Qwen2-72B-Instruct | To find the remainder when the $100^{\mathrm{th}}$ term of the Fibonacci sequence is divided by 4, we can look at the pattern of remainders when Fibonacci numbers are divided by 4. The Fibonacci sequence modulo 4 begins as follows: 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, ... The pattern repeats every 6 terms. Therefore, to find the remainder of the $100^{\mathrm{th}}$ term when divided by 4, we can find the remainder of 100 when divided by 6, which tells us where in the repeating pattern of 6 the $100^{\mathrm{th}}$ term falls. $100 \mod 6 = 4$ This means the $100^{\mathrm{th}}$ term has the same remainder when divided by 4 as the 4th term in the repeating pattern. Looking at the pattern, the 4th term is 3. Answer: B. 3 |
B |
| baichuan4 | B | B |
| doubao | B The Fibonacci sequence modulo 4 has a period of 6: 1, 1, 2, 3, 1, 0. 100 ÷ 6 = 16 R 4, and the fourth number in the period is 3. |
B |
| minimax | The Fibonacci sequence modulo 4 has a repeating pattern. Let's look at the first few terms of the sequence modulo 4: 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, ... Notice that after the first four terms, the sequence starts repeating the pattern (1, 1, 2, 3). This is because: - 1 + 1 = 2 (mod 4) - 1 + 2 = 3 (mod 4) - 2 + 3 = 5 ≡ 1 (mod 4) - 3 + 1 = 4 ≡ 0 (mod 4) - 1 + 0 = 1 (mod 4) - 0 + 1 = 1 (mod 4) - 1 + 1 = 2 (mod 4) - 1 + 2 = 3 (mod 4) ... and so on. Since the sequence repeats every 4 terms, to find the remainder of the 100th term when divided by 4, we can simply find the remainder of 100 when divided by 4, because the sequence will have completed 25 full cycles of the pattern (1, 1, 2, 3) and will be at the same point in the cycle as the 4th term. 100 divided by 4 gives a quotient of 25 with a remainder of 0. Therefore, the 100th term of the Fibonacci sequence will have the same remainder as the 4th term when divided by 4, which is 3. The answer is B. 3. |
B |
| kimi | To find the remainder when the 100th term of the Fibonacci sequence is divided by 4, we can look at the pattern of the Fibonacci sequence modulo 4. The Fibonacci sequence modulo 4 is: 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, ... Notice that the sequence repeats every 6 terms: 1, 1, 2, 3, 1, 0. To find the 100th term modulo 4, we can divide 100 by 6 and find the remainder: 100 ÷ 6 = 16 with a remainder of 4. So, the 100th term in the Fibonacci sequence will have the same remainder when divided by 4 as the 4th term in the repeating sequence. The 4th term in the sequence 1, 1, 2, 3, 1, 0 is 3. Therefore, the remainder when the 100th term of the Fibonacci sequence is divided by 4 is 3. Answer: B. 3 |
B |
| wenxinyiyan | B Explanation: The Fibonacci sequence is defined by the recursion $F_n = F_{n-1} + F_{n-2}$, with $F_1 = F_2 = 1$. We are interested in finding the remainder when the $100^{\text{th}}$ term is divided by 4. Let's compute the first few terms modulo 4: * $F_1 \equiv 1 \pmod{4}$ * $F_2 \equiv 1 \pmod{4}$ * $F_3 = F_1 + F_2 = 1 + 1 = 2 \Rightarrow F_3 \equiv 2 \pmod{4}$ * $F_4 = F_2 + F_3 = 1 + 2 = 3 \Rightarrow F_4 \equiv 3 \pmod{4}$ * $F_5 = F_3 + F_4 = 2 + 3 = 5 \Rightarrow F_5 \equiv 1 \pmod{4}$ * $F_6 = F_4 + F_5 = 3 + 5 = 8 \Rightarrow F_6 \equiv 0 \pmod{4}$ * $F_7 = F_5 + F_6 = 1 + 8 = 9 \Rightarrow F_7 \equiv 1 \pmod{4}$ * $F_8 = F_6 + F_7 = 0 + 9 = 9 \Rightarrow F_8 \equiv 1 \pmod{4}$ Observe that the sequence of remainders starts to repeat: 1, 1, 2, 3, 1, 0, 1, 1, ... It seems that there is a pattern of length 6 (1, 1, 2, 3, 1, 0) that repeats indefinitely. To find the remainder of the $100^{\text{th}}$ term, we can compute $100 \mod 6 = 4$. Therefore, the remainder of the $100^{\text{th}}$ term of the Fibonacci sequence modulo 4 will be the same as the $4^{\text{th}}$ term in the pattern of remainders, which is 3. |
B |
| zhipu | To determine the remainder when the \(100^{\text{th}}\) term of the Fibonacci sequence is divided by 4, we need to understand the pattern of the Fibonacci sequence modulo 4. Let's start by listing the Fibonacci sequence modulo 4: \[ \begin{align*} F_1 & = 1 \equiv 1 \pmod{4}, \\ F_2 & = 1 \equiv 1 \pmod{4}, \\ F_3 & = 2 \equiv 2 \pmod{4}, \\ F_4 & = 3 \equiv 3 \pmod{4}, \\ F_5 & = 5 \equiv 1 \pmod{4}, \\ F_6 & = 8 \equiv 0 \pmod{4}, \\ F_7 & = 13 \equiv 1 \pmod{4}, \\ F_8 & = 21 \equiv 1 \pmod{4}, \\ F_9 & = 34 \equiv 2 \pmod{4}, \\ F_{10} & = 55 \equiv 3 \pmod{4}, \\ F_{11} & = 89 \equiv 1 \pmod{4}, \\ F_{12} & = 144 \equiv 0 \pmod{4}. \end{align*} \] From this, we observe the repeating pattern: \[ 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, \ldots \] The pattern repeats every 6 terms. To find the remainder of the \(100^{\text{th}}\) term modulo 4, we need to determine the position of \(F_{100}\) within this repeating cycle. We compute: \[ 100 \mod 6 = 4 \] This means that the \(100^{\text{th}}\) term of the Fibonacci sequence corresponds to the 4th term in the repeating pattern. From our pattern list: \[ 1, 1, 2, 3, 1, 0, \ldots \] The 4th term in the pattern is \(3\). Therefore, the remainder when the \(100^{\text{th}}\) term of the Fibonacci sequence is divided by 4 is \(3\). The correct answer is: \[ \boxed{B} \] |
B |
| gpto1 | Answer: B | B |
| deepseekv3 | server error: Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-994b71eb-bf4c-916c-a4e6-ca5e7c890f9d', 'request_id': '994b71eb-bf4c-916c-a4e6-ca5e7c890f9d'} | B |