金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have \[ d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .\]
-Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have \[ d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .\]

The solution is:
We can start by verifying that $n=3$ and $n=4$ work by listing out the factors of $3!$ and $4!$ . We can also see that $n=5$ does not work because the terms $15, 20$ , and $24$ are consecutive factors of $5!$ . Also, $n=6$ does not work because the terms $6, 8$ , and $9$ appear consecutively in the factors of <missing 1> .
-We can start by verifying that $n=3$ and $n=4$ work by listing out the factors of $3!$ and $4!$ . We can also see that $n=5$ does not work because the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> . Also, $n=6$ does not work because the terms $6, 8$ , and $9$ appear consecutively in the factors of $6!$ .

Note that if we have a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$ , then the condition cannot be satisfied.

If $n\geq7$ is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$ . Also, $(n-2)(n)=n^2-2n$ is a factor of $n!$ . Since $2n<n^2-2n$ for all $n\geq7$ , we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that $n<p<n^2-2n$ . Since we have two consecutive factors of $n!$ and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all odd $n\geq7$ .

If $n\geq8$ is even, then $(2)(\frac{n-2}{2})(n-2)=n^2-4n+4$ is a factor of $n!$ . Also, $(n-3)(n-1)=n^2-4n+3$ is a factor of $n!$ . Since $2n<n^2-4n+3$ for all $n\geq8$ , we can use Bertrand's Postulate again to show that there is at least one prime number $p$ such that $n<p<n^2-4n+3$ . Since we have two consecutive factors of $n!$ and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all even $n\geq8$ .

Therefore, the only numbers that work are <missing 4> and $n=4$ .

The formulae are:
<expression 1> 5!
<expression 2> n=3
<expression 3> p>n
<expression 4> 6!

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	<Detailed reasoning> To match the formulae to the missing tags in the solution, we need to identify which expressions correspond to the descriptions provided in the solution text. 1. <missing 1>: The text mentions "the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>". This refers to the factors of a specific factorial, and since the example given is for $n=6$, it implies that <missing 1> should be the factorial of 6, which is $6!$. Therefore, <missing 1> corresponds to <expression 4>. 2. <missing 2>: The text states "the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This refers to the factors of $5!$, as these numbers are factors of 120 (which is $5!$). Therefore, <missing 2> corresponds to <expression 1>. 3. <missing 3>: The text mentions "a prime number <missing 3>". This refers to a prime number greater than $n$, which is represented by $p > n$. Therefore, <missing 3> corresponds to <expression 3>. 4. <missing 4>: The text concludes with "the only numbers that work are <missing 4> and $n=4$". This refers to the integers $n$ that satisfy the condition, and from the solution, we know that $n=3$ works. Therefore, <missing 4> corresponds to <expression 2>. Answer: 4, 1, 3, 2	4,1,3,2
deepseek-r1	Answer: 4,1,3,2 Detailed reasoning: 1. <missing 1> refers to the factorial of 6 (n=6), so it's 6! (expression 4). 2. <missing 2> discusses n=5's factorial, which is 5! (expression 1). 3. <missing 3> requires a prime greater than n, matching p>n (expression 3). 4. <missing 4> lists valid solutions, starting with n=3 (expression 2). Answer: 4,1,3,2	4,1,3,2
gpto1	Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}	4,1,3,2
gpt-4o-2024-08-06	To solve this problem, we need to fill in the missing parts of the solution with the correct expressions. Let's go through each missing part step by step. 1. <missing 1>: The text mentions that $n=6$ does not work because the terms $6, 8$, and $9$ appear consecutively in the factors of a certain factorial. Since $n=6$, the factorial in question is $6!$. Therefore, <missing 1> should be filled with the expression for $6!$. 2. <missing 2>: The text states that $n=5$ does not work because the terms $15, 20$, and $24$ are consecutive factors of a certain factorial. Since $n=5$, the factorial in question is $5!$. Therefore, <missing 2> should be filled with the expression for $5!$. 3. <missing 3>: The text mentions a prime number and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$. The condition is that there is a prime number greater than $n$. Therefore, <missing 3> should be filled with the expression for $p>n$. 4. <missing 4>: The text concludes that the only numbers that work are $n=3$ and $n=4$. Therefore, <missing 4> should be filled with the expression for $n=3$. Now, let's match these conclusions with the given expressions: - <missing 1> corresponds to $6!$, which is <expression 4>. - <missing 2> corresponds to $5!$, which is <expression 1>. - <missing 3> corresponds to $p>n$, which is <expression 3>. - <missing 4> corresponds to $n=3$, which is <expression 2>. Therefore, the final answer is: Answer: 4, 1, 3, 2	4,1,3,2
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> Answer: 1, 4, 3, 2	4,1,3,2
Meta-Llama-3.1-405B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. The first missing tag is <missing 1>, which refers to the factors of a certain number. The correct formula to fill this tag is <expression 4>, which is 6!. The second missing tag is <missing 2>, which also refers to the factors of a certain number. The correct formula to fill this tag is <expression 1>, which is 5!. The third missing tag is <missing 3>, which refers to a prime number. The correct formula to fill this tag is <expression 3>, which is p>n. The fourth missing tag is <missing 4>, which refers to the numbers that work. The correct formula to fill this tag is <expression 2>, which is n=3. Therefore, the correct answer is: Answer: 4, 1, 3, 2	4,1,3,2
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. First, let's analyze the solution and identify the missing tags: 1. "...the factors of <missing 1> ." 2. "...the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> ." 3. "...a prime number <missing 3> and an integer $k>p$..." 4. "...the only numbers that work are <missing 4> and $n=4$ ." Now, let's examine the formulae: 1. <expression 1> 5! 2. <expression 2> n=3 3. <expression 3> p>n 4. <expression 4> 6! Based on the context, we can make the following matches: 1. "...the factors of <missing 1> ." seems to be related to the factors of 6!, so <missing 1> should be <expression 4> 6!. 2. "...the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> ." seems to be related to the factors of 5!, so <missing 2> should be <expression 1> 5!. 3. "...a prime number <missing 3> and an integer $k>p$..." seems to be related to the inequality p>n, so <missing 3> should be <expression 3> p>n. 4. "...the only numbers that work are <missing 4> and $n=4$ ." seems to be related to the value n=3, so <missing 4> should be <expression 2> n=3. Therefore, the answer is: Answer: 4, 1, 3, 2	4,1,3,2
Qwen2-72B-Instruct	The first missing part refers to the factorial of 6, which is represented by expression 4. The second missing part also refers to the factorial of 5, which is represented by expression 1. The third missing part refers to a prime number greater than n, which is represented by expression 3. The fourth missing part refers to the integer solutions, starting with n=3, which is represented by expression 2. Answer: 4, 1, 3, 2	4,1,3,2
baichuan4	Detailed reasoning: 1. The first missing tag is in the context of discussing the factors of $n!$ when $n=6$. The text mentions that the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>. Given that the discussion is about $n=6$, it is clear that the missing tag should refer to $6!$. Therefore, <missing 1> corresponds to <expression 4>. 2. The second missing tag is in the context of discussing why $n=5$ does not work. It mentions that the terms $15, 20$, and $24$ are consecutive factors of <missing 2>. Since the discussion is about $n=5$, it is clear that the missing tag should refer to $5!$. Therefore, <missing 2> corresponds to <expression 1>. 3. The third missing tag is in the context of discussing a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$. The text implies that $p$ is a prime number greater than $n$. Therefore, <missing 3> corresponds to <expression 3>. 4. The fourth missing tag is in the conclusion, stating that the only numbers that work are <missing 4> and $n=4$. Given that the discussion has been about $n \geq 3$, and earlier it was verified that $n=3$ works, it is clear that <missing 4> should refer to $n=3$. Therefore, <missing 4> corresponds to <expression 2>. Answer: 4, 1, 3, 2	4,1,3,2
doubao	<Detailed reasoning>: In the solution, "the terms 6, 8, and 9 appear consecutively in the factors of" should be filled with <expression 4> which is 6!, and "the terms 15, 20, and 24 are consecutive factors of" should be filled with <expression 1> which is 5!. Also, "If $n\geq7$ is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$" and other similar statements about factors of $n!$ refer to expressions based on $n$, not the specific given expressions. Finally, "the only numbers that work are" should be filled with <expression 2> which is n=3. Answer: 1, 4, 2	4,1,3,2
gpto1	Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}	4,1,3,2
kimi	<Detailed reasoning> 1. The first missing tag <missing 1> is referring to the factorial of a number, and since we are discussing the factors of $n!$, the only factorial that fits the context is $6!$. So, the first missing tag is filled with expression 4. 2. The second missing tag <missing 2> is referring to a factorial that has consecutive factors $15, 20$, and $24$. The only factorial that fits this description is $5!$. So, the second missing tag is filled with expression 1. 3. The third missing tag <missing 3> is referring to a prime number greater than some integer $n$. The only expression that fits this description is $p>n$. So, the third missing tag is filled with expression 3. 4. The fourth missing tag <missing 4> is referring to the only numbers that work for the given property. Since we have already established that $n=3$ and $n=4$ work, the only expression that fits this description is $n=3$. So, the fourth missing tag is filled with expression 2. Answer: 4, 1, 3, 2	4,1,3,2
minimax	To solve this problem, we need to match the missing parts of the solution with the corresponding expressions provided. First, let's look at the missing parts in the solution: 1. <missing 1> should be the factorial of a number that is used to show that $n=6$ does not work. The correct factorial here should be $6!$ because the solution mentions the terms $6, 8$, and $9$ appear consecutively in the factors of something, and since we are talking about $n=6$, it should be $6!$. 2. <missing 2> is used in the context of $n=5$ not working because of consecutive factors of $5!$. The correct factorial here should be $5!$. 3. <missing 3> is a prime number that is used to show that if there is a prime number between two consecutive factors of $n!$, the condition cannot be satisfied. The correct expression here should be $p>n$ because the solution is discussing a prime number greater than $n$. 4. <missing 4> is the conclusion about which numbers work. The solution states that the only numbers that work are some specific numbers and $n=4$. Since	4,1,3,2
wenxinyiyan	Detailed reasoning: 1. The first missing tag <missing 1> refers to the factors of 6!. This is because the context mentions that $n=6$ does not work due to the consecutive factors 6, 8, and 9 in 6!. Therefore, <missing 1> should be replaced by <expression 4>. 2. The second missing tag <missing 2> is in the context of the factors of 5!. The text says that $n=5$ does not work because of the consecutive factors 15, 20, and 24 in 5!. Hence, <missing 2> should be replaced by <expression 1>. 3. The third missing tag <missing 3> talks about a prime number p greater than n. The condition mentioned is that if there are consecutive factors k and k+1 of n! such that p is between k and n, then the condition cannot be satisfied. This refers to <expression 3>. 4. The last missing tag <missing 4> is in the conclusion part of the solution where it says that the only numbers that work are n=3 and n=4. Since n=3 is already mentioned, <missing 4> should be replaced by <expression 2>. Answer: 4, 1, 3, 2	4,1,3,2
zhipu	### Detailed Reasoning 1. Identifying <missing 1>: - The context around <missing 1> mentions "the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>". This implies we are talking about the factors of $6!$. - From the given formulae, <expression 4> corresponds to $6!$. - Therefore, <missing 1> should be filled by <expression 4>. 2. Identifying <missing 2>: - The context around <missing 2> mentions "the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This implies we are talking about the factors of $5!$. - From the given formulae, <expression 1> corresponds to $5!$. - Therefore, <missing 2> should be filled by <expression 1>. 3. Identifying <missing 3>: - The context around <missing 3> mentions "if we have a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$, then the condition cannot be satisfied." - This implies <missing 3> should be a placeholder for a prime number greater than $n$. - From the given formulae, <expression 3> corresponds to $p>n$. - Therefore, <missing 3> should be filled by <expression 3>. 4. Identifying <missing 4>: - The context around <missing 4> mentions "Therefore, the only numbers that work are <missing 4> and $n=4$." - This implies <missing 4> should be a placeholder for another specific value of $n$ that works, which is $n=3$. - From the given formulae, <expression 2> corresponds to $n=3$. - Therefore, <missing 4> should be filled by <expression 2>. ### Answer 4, 1, 3, 2	4,1,3,2
deepseek-v3	<Detailed reasoning> 1. <missing 1>: The solution mentions that "the terms \(6, 8\), and \(9\) appear consecutively in the factors of <missing 1>." Since the context is about \(n=6\), the factorial in question should be \(6!\). Therefore, <missing 1> should be filled with <expression 4>. 2. <missing 2>: The solution discusses \(n=5\) and mentions that "the terms \(15, 20\), and \(24\) are consecutive factors of <missing 2>." Here, the factorial in question is \(5!\). Therefore, <missing 2> should be filled with <expression 1>. 3. <missing 3>: The solution states that "if we have a prime number <missing 3> and an integer \(k>p\) such that both \(k\) and \(k+1\) are factors of \(n!\), then the condition cannot be satisfied." The condition here is that the prime number must be greater than \(n\). Therefore, <missing 3> should be filled with <expression 3>. 4. <missing 4>: The solution concludes that "the only numbers that work are <missing 4> and \(n=4\)." From the earlier discussion, we know that \(n=3\) also works. Therefore, <missing 4> should be filled with <expression 2>. Answer: 4, 1, 3, 2	4,1,3,2

模型

模型答案

正确答案

金声玉亮2.0

<Detailed reasoning>

To match the formulae to the missing tags in the solution, we need to identify which expressions correspond to the descriptions provided in the solution text.

1. **<missing 1>**: The text mentions "the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>". This refers to the factors of a specific factorial, and since the example given is for $n=6$, it implies that <missing 1> should be the factorial of 6, which is $6!$. Therefore, <missing 1> corresponds to <expression 4>.

2. **<missing 2>**: The text states "the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This refers to the factors of $5!$, as these numbers are factors of 120 (which is $5!$). Therefore, <missing 2> corresponds to <expression 1>.

3. **<missing 3>**: The text mentions "a prime number <missing 3>". This refers to a prime number greater than $n$, which is represented by $p > n$. Therefore, <missing 3> corresponds to <expression 3>.

4. **<missing 4>**: The text concludes with "the only numbers that work are <missing 4> and $n=4$". This refers to the integers $n$ that satisfy the condition, and from the solution, we know that $n=3$ works. Therefore, <missing 4> corresponds to <expression 2>.

Answer: 4, 1, 3, 2

4,1,3,2

deepseek-r1

Answer: 4,1,3,2

Detailed reasoning:

1. **<missing 1>** refers to the factorial of 6 (n=6), so it's **6!** (expression 4).
2. **<missing 2>** discusses n=5's factorial, which is **5!** (expression 1).
3. **<missing 3>** requires a prime greater than n, matching **p>n** (expression 3).
4. **<missing 4>** lists valid solutions, starting with **n=3** (expression 2).

Answer: 4,1,3,2

4,1,3,2

gpto1

Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}

4,1,3,2

gpt-4o-2024-08-06

To solve this problem, we need to fill in the missing parts of the solution with the correct expressions. Let's go through each missing part step by step.

1. **<missing 1>**: The text mentions that $n=6$ does not work because the terms $6, 8$, and $9$ appear consecutively in the factors of a certain factorial. Since $n=6$, the factorial in question is $6!$. Therefore, <missing 1> should be filled with the expression for $6!$.

2. **<missing 2>**: The text states that $n=5$ does not work because the terms $15, 20$, and $24$ are consecutive factors of a certain factorial. Since $n=5$, the factorial in question is $5!$. Therefore, <missing 2> should be filled with the expression for $5!$.

3. **<missing 3>**: The text mentions a prime number and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$. The condition is that there is a prime number greater than $n$. Therefore, <missing 3> should be filled with the expression for $p>n$.

4. **<missing 4>**: The text concludes that the only numbers that work are $n=3$ and $n=4$. Therefore, <missing 4> should be filled with the expression for $n=3$.

Now, let's match these conclusions with the given expressions:

- **<missing 1>** corresponds to $6!$, which is <expression 4>.
- **<missing 2>** corresponds to $5!$, which is <expression 1>.
- **<missing 3>** corresponds to $p>n$, which is <expression 3>.
- **<missing 4>** corresponds to $n=3$, which is <expression 2>.

Therefore, the final answer is:

Answer: 4, 1, 3, 2

4,1,3,2

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

Answer: 1, 4, 3, 2

4,1,3,2

Meta-Llama-3.1-405B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution.

The first missing tag is <missing 1>, which refers to the factors of a certain number. The correct formula to fill this tag is <expression 4>, which is 6!.

The second missing tag is <missing 2>, which also refers to the factors of a certain number. The correct formula to fill this tag is <expression 1>, which is 5!.

The third missing tag is <missing 3>, which refers to a prime number. The correct formula to fill this tag is <expression 3>, which is p>n.

The fourth missing tag is <missing 4>, which refers to the numbers that work. The correct formula to fill this tag is <expression 2>, which is n=3.

Therefore, the correct answer is:

Answer: 4, 1, 3, 2

4,1,3,2

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution.

First, let's analyze the solution and identify the missing tags:

1. "...the factors of <missing 1> ."
2. "...the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> ."
3. "...a prime number <missing 3> and an integer $k>p$..."
4. "...the only numbers that work are <missing 4> and $n=4$ ."

Now, let's examine the formulae:

1. <expression 1> 5!
2. <expression 2> n=3
3. <expression 3> p>n
4. <expression 4> 6!

Based on the context, we can make the following matches:

1. "...the factors of <missing 1> ." seems to be related to the factors of 6!, so <missing 1> should be <expression 4> 6!.
2. "...the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> ." seems to be related to the factors of 5!, so <missing 2> should be <expression 1> 5!.
3. "...a prime number <missing 3> and an integer $k>p$..." seems to be related to the inequality p>n, so <missing 3> should be <expression 3> p>n.
4. "...the only numbers that work are <missing 4> and $n=4$ ." seems to be related to the value n=3, so <missing 4> should be <expression 2> n=3.

Therefore, the answer is:

Answer: 4, 1, 3, 2

4,1,3,2

Qwen2-72B-Instruct

The first missing part refers to the factorial of 6, which is represented by expression 4.
The second missing part also refers to the factorial of 5, which is represented by expression 1.
The third missing part refers to a prime number greater than n, which is represented by expression 3.
The fourth missing part refers to the integer solutions, starting with n=3, which is represented by expression 2.

Answer: 4, 1, 3, 2

4,1,3,2

baichuan4

Detailed reasoning:

1. The first missing tag is in the context of discussing the factors of $n!$ when $n=6$. The text mentions that the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>. Given that the discussion is about $n=6$, it is clear that the missing tag should refer to $6!$. Therefore, <missing 1> corresponds to <expression 4>.

2. The second missing tag is in the context of discussing why $n=5$ does not work. It mentions that the terms $15, 20$, and $24$ are consecutive factors of <missing 2>. Since the discussion is about $n=5$, it is clear that the missing tag should refer to $5!$. Therefore, <missing 2> corresponds to <expression 1>.

3. The third missing tag is in the context of discussing a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$. The text implies that $p$ is a prime number greater than $n$. Therefore, <missing 3> corresponds to <expression 3>.

4. The fourth missing tag is in the conclusion, stating that the only numbers that work are <missing 4> and $n=4$. Given that the discussion has been about $n \geq 3$, and earlier it was verified that $n=3$ works, it is clear that <missing 4> should refer to $n=3$. Therefore, <missing 4> corresponds to <expression 2>.

Answer: 4, 1, 3, 2

4,1,3,2

doubao

<Detailed reasoning>: In the solution, "the terms 6, 8, and 9 appear consecutively in the factors of" should be filled with <expression 4> which is 6!, and "the terms 15, 20, and 24 are consecutive factors of" should be filled with <expression 1> which is 5!. Also, "If $n\geq7$ is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$" and other similar statements about factors of $n!$ refer to expressions based on $n$, not the specific given expressions. Finally, "the only numbers that work are" should be filled with <expression 2> which is n=3.
Answer: 1, 4, 2

4,1,3,2

gpto1

4,1,3,2

kimi

<Detailed reasoning>

1. The first missing tag <missing 1> is referring to the factorial of a number, and since we are discussing the factors of $n!$, the only factorial that fits the context is $6!$. So, the first missing tag is filled with expression 4.
2. The second missing tag <missing 2> is referring to a factorial that has consecutive factors $15, 20$, and $24$. The only factorial that fits this description is $5!$. So, the second missing tag is filled with expression 1.
3. The third missing tag <missing 3> is referring to a prime number greater than some integer $n$. The only expression that fits this description is $p>n$. So, the third missing tag is filled with expression 3.
4. The fourth missing tag <missing 4> is referring to the only numbers that work for the given property. Since we have already established that $n=3$ and $n=4$ work, the only expression that fits this description is $n=3$. So, the fourth missing tag is filled with expression 2.

Answer: 4, 1, 3, 2

4,1,3,2

minimax

To solve this problem, we need to match the missing parts of the solution with the corresponding expressions provided.

First, let's look at the missing parts in the solution:

1. <missing 1> should be the factorial of a number that is used to show that $n=6$ does not work. The correct factorial here should be $6!$ because the solution mentions the terms $6, 8$, and $9$ appear consecutively in the factors of something, and since we are talking about $n=6$, it should be $6!$.

2. <missing 2> is used in the context of $n=5$ not working because of consecutive factors of $5!$. The correct factorial here should be $5!$.

3. <missing 3> is a prime number that is used to show that if there is a prime number between two consecutive factors of $n!$, the condition cannot be satisfied. The correct expression here should be $p>n$ because the solution is discussing a prime number greater than $n$.

4. <missing 4> is the conclusion about which numbers work. The solution states that the only numbers that work are some specific numbers and $n=4$. Since

4,1,3,2

wenxinyiyan

Detailed reasoning:

1. The first missing tag <missing 1> refers to the factors of 6!. This is because the context mentions that $n=6$ does not work due to the consecutive factors 6, 8, and 9 in 6!. Therefore, <missing 1> should be replaced by <expression 4>.

2. The second missing tag <missing 2> is in the context of the factors of 5!. The text says that $n=5$ does not work because of the consecutive factors 15, 20, and 24 in 5!. Hence, <missing 2> should be replaced by <expression 1>.

3. The third missing tag <missing 3> talks about a prime number p greater than n. The condition mentioned is that if there are consecutive factors k and k+1 of n! such that p is between k and n, then the condition cannot be satisfied. This refers to <expression 3>.

4. The last missing tag <missing 4> is in the conclusion part of the solution where it says that the only numbers that work are n=3 and n=4. Since n=3 is already mentioned, <missing 4> should be replaced by <expression 2>.

Answer: 4, 1, 3, 2

4,1,3,2

zhipu

### Detailed Reasoning

1. **Identifying <missing 1>**:
- The context around <missing 1> mentions "the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>". This implies we are talking about the factors of $6!$.
- From the given formulae, <expression 4> corresponds to $6!$.
- Therefore, <missing 1> should be filled by <expression 4>.

2. **Identifying <missing 2>**:
- The context around <missing 2> mentions "the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This implies we are talking about the factors of $5!$.
- From the given formulae, <expression 1> corresponds to $5!$.
- Therefore, <missing 2> should be filled by <expression 1>.

3. **Identifying <missing 3>**:
- The context around <missing 3> mentions "if we have a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$, then the condition cannot be satisfied."
- This implies <missing 3> should be a placeholder for a prime number greater than $n$.
- From the given formulae, <expression 3> corresponds to $p>n$.
- Therefore, <missing 3> should be filled by <expression 3>.

4. **Identifying <missing 4>**:
- The context around <missing 4> mentions "Therefore, the only numbers that work are <missing 4> and $n=4$."
- This implies <missing 4> should be a placeholder for another specific value of $n$ that works, which is $n=3$.
- From the given formulae, <expression 2> corresponds to $n=3$.
- Therefore, <missing 4> should be filled by <expression 2>.

### Answer
4, 1, 3, 2

4,1,3,2

deepseek-v3

<Detailed reasoning>

1. **<missing 1>**: The solution mentions that "the terms $6, 8$, and $9$ appear consecutively in the factors of <missing 1>." Since the context is about $n=6$, the factorial in question should be $6!$. Therefore, <missing 1> should be filled with **<expression 4>**.

2. **<missing 2>**: The solution discusses $n=5$ and mentions that "the terms $15, 20$, and $24$ are consecutive factors of <missing 2>." Here, the factorial in question is $5!$. Therefore, <missing 2> should be filled with **<expression 1>**.

3. **<missing 3>**: The solution states that "if we have a prime number <missing 3> and an integer $k>p$ such that both $k$ and $k+1$ are factors of $n!$, then the condition cannot be satisfied." The condition here is that the prime number must be greater than $n$. Therefore, <missing 3> should be filled with **<expression 3>**.

4. **<missing 4>**: The solution concludes that "the only numbers that work are <missing 4> and $n=4$." From the earlier discussion, we know that $n=3$ also works. Therefore, <missing 4> should be filled with **<expression 2>**.

Answer: 4, 1, 3, 2

4,1,3,2