A+B Problem (Case Count)

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本文介绍了一个简单的控制台应用程序，该程序使用C++实现了一个基本的加法计算器功能。用户可以输入多个成对的整数，程序将输出每一对整数的和。此程序通过循环读取输入并利用标准I/O流完成计算。
部署运行你感兴趣的模型镜像

代码：

#include <iostream>
using namespace std;
int main()
{
    int a,b,n;
    cin >> n;             
    while(n)
    {
        cin>> a>>b;
        cout<<a + b<<endl;
        n--;
    }
    return 0;
}



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{"id": "cmpl-e759f3d8-78e1-11f0-b2b4-1a85109d8f07", "choices": [{"finish_reason": "stop", "index": 0, "message": {"content": "<think>\nOkay, so I need to figure out how many whole-number divisors there are for the integer given by the expression $ a(2a + b) - 2a^2 + ab $, where $ a $ and $ b $ are different prime numbers greater than 2. Let me start by simplifying the expression step by step.\n\nFirst, let me write down the expression again to make sure I have it right: $ a(2a + b) - 2a^2 + ab $. Let me expand the first term, which is $ a(2a + b) $. If I distribute the $ a $, that becomes $ 2a^2 + ab $. So substituting that back into the original expression, we have:\n\n$ 2a^2 + ab - 2a^2 + ab $.\n\nNow let me combine like terms. The $ 2a^2 $ and $ -2a^2 $ cancel each other out, right? So that leaves me with $ ab + ab $, which is $ 2ab $. So the entire expression simplifies to $ 2ab $. \n\nWait, so the problem is asking for the number of whole-number divisors of $ 2ab $, where $ a $ and $ b $ are different primes greater than 2. Let me confirm that I simplified correctly. Let me check again:\n\nOriginal expression: $ a(2a + b) - 2a^2 + ab $.\n\nExpanding $ a(2a + b) $: $ 2a^2 + ab $. Then subtract $ 2a^2 $, so $ 2a^2 + ab - 2a^2 = ab $. Then add $ ab $, so $ ab + ab = 2ab $. Yes, that seems right. So the expression simplifies to $ 2ab $.\n\nAlright, so now I need to find the number of positive divisors of $ 2ab $. Since $ a $ and $ b $ are primes greater than 2, they are both odd primes. Let me recall that the number of divisors of a number is found by taking its prime factorization, adding one to each of the exponents, and multiplying those together.\n\nSo first, let me write the prime factorization of $ 2ab $. Since 2 is a prime number, and $ a $ and $ b $ are different primes, all three primes are distinct. Therefore, the prime factors of $ 2ab $ are 2, $ a $, and $ b $. Each of these primes has an exponent of 1 in the factorization. So the prime factorization is $ 2^1 \\times a^1 \\times b^1 $.\n\nTo find the number of divisors, I take each exponent (which are all 1s), add 1 to them, and multiply. That would be $ (1 + 1)(1 + 1)(1 + 1) = 2 \\times 2 \\times 2 = 8 $. Therefore, there are 8 positive divisors.\n\nBut let me verify this with an example to make sure I haven't made a mistake. Let me pick specific primes for $ a $ and $ b $. Let's take $ a = 3 $ and $ b = 5 $, which are both primes greater than 2 and different.\n\nCalculating $ 2ab $: $ 2 \\times 3 \\times 5 = 30 $. Now, the divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. That's 8 divisors. Yep, that matches the calculation. Let me try another example to be safe. Let me take $ a = 5 $, $ b = 7 $. Then $ 2ab = 2 \\times 5 \\times 7 = 70 $. Divisors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Again, 8 divisors. Okay, so that seems consistent.\n\nBut let me think again if there's any possibility that 2 could be equal to $ a $ or $ b $. However, the problem states that $ a $ and $ b $ are primes greater than 2, so they are odd primes. Therefore, 2 is not among them. Therefore, 2, $ a $, and $ b $ are distinct primes. Therefore, the prime factorization is indeed three distinct primes each to the first power, leading to 2×2×2=8 divisors.\n\nBut let me check if there's a case where maybe $ a $ or $ b $ is 2. But the problem says they are greater than 2, so they can't be. Therefore, my reasoning holds.\n\nTherefore, the number of whole-number divisors is 8. Therefore, the answer is 8, which in boxed form is \\boxed{8}.\n\nWait, but let me just check once more. Suppose someone might think that maybe the original expression simplifies differently? Let me check again:\n\nOriginal expression: $ a(2a + b) - 2a^2 + ab $.\n\nFirst term: $ a(2a + b) = 2a^2 + ab $.\n\nSubtracting $ 2a^2 $: $ 2a^2 + ab - 2a^2 = ab $.\n\nAdding $ ab $: $ ab + ab = 2ab $. Yes, that's correct. So the expression simplifies to 2ab. So no issues there.\n\nAlternatively, maybe someone might misread the original problem? Let me check again. The problem says: \"the integer $ a(2a + b) - 2a^2 + ab $\". So yes, that's how it's written. So the steps are correct.\n\nTherefore, I think my answer is correct. The number of divisors is 8.\n\n**Final Answer**\n\\boxed{8}\n</think>\n\nWe are given an expression involving two distinct prime numbers $ a $ and $ b $, both greater than 2, and asked to find the number of whole-number divisors of the integer:\n\n$$\na(2a + b) - 2a^2 + ab\n$$\n\n---\n\n### Step 1: Simplify the Expression\n\nStart by expanding the first term:\n\n$$\na(2a + b) = 2a^2 + ab\n$$\n\nNow substitute this back into the full expression:\n\n$$\n2a^2 + ab - 2a^2 + ab\n$$\n\nSimplify by combining like terms:\n\n- $ 2a^2 - 2a^2 = 0 $\n- $ ab + ab = 2ab $\n\nSo the entire expression simplifies to:\n\n$$\n2ab\n$$\n\n---\n\n### Step 2: Prime Factorization of $ 2ab $\n\nSince $ a $ and $ b $ are distinct prime numbers greater than 2, and 2 is also a prime, the expression $ 2ab $ is the product of three **distinct** primes:\n\n$$\n2ab = 2^1 \\cdot a^1 \\cdot b^1\n$$\n\n---\n\n### Step 3: Count the Number of Divisors\n\nThe number of positive divisors of a number is determined by taking the product of one more than each exponent in its prime factorization.\n\nHere, the exponents are all 1:\n\n$$\n(1 + 1)(1 + 1)(1 + 1) = 2 \\cdot 2 \\cdot 2 = 8\n$$\n\n---\n\n### Final Answer\n\n$$\n\\boxed{8}\n$$", "role": "assistant", "reasoning_content": ""}, "review": {"gold": "8", "pred": "8", "result": true}}], "created": 1755157209, "model": "Qwen3-32B", "object": "chat.completion", "usage": {"completion_tokens": 1627, "prompt_tokens": 68, "total_tokens": 1695}, "model_spec": {"api_url": "http://0.0.0.0:9009/v1/chat/completions", "model_id": "Qwen3-32B", "api_key": "EMPTY"}, "answer_id": "answer-1a07ae97260436f698e63d4620d7cff0", "subset_name": "default", "raw_input": {"problem": "Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\cdot a^1\\cdot b^1$ has $(1+1)(1+1)(1+1)=\\boxed{8}$ divisors.", "answer": "8", "subject": "Number Theory", "level": 4, "unique_id": "test/number_theory/1287.json"}, "origin_prompt": {"data": ["Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?\nPlease reason step by step, and put your final answer within \\boxed{}."], "system_prompt": null, "raw_input": {"problem": "Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\cdot a^1\\cdot b^1$ has $(1+1)(1+1)(1+1)=\\boxed{8}$ divisors.", "answer": "8", "subject": "Number Theory", "level": 4, "unique_id": "test/number_theory/1287.json"}, "index": 0}, "reviewed": true, "review_id": "review-307a3dc16716316d92e62e1f3ae09703", "reviewer_spec": {"metric": ["AveragePass@1"], "reviewer": ["Evaluator"], "revision": ["default"]}, "review_time": 1755194244.6042051} 写一个脚本，将以上字符串格式化为以下json格式 { "id": "cmpl-e759f3d8-78e1-11f0-b2b4-1a85109d8f07", "choices": [ { "finish_reason": "stop", "index": 0, "message": { "content": "<think>\nOkay, so I need to figure out how many whole-number divisors there are for the integer given by the expression $ a(2a + b) - 2a^2 + ab $, where $ a $ and $ b $ are different prime numbers greater than 2. Let me start by simplifying the expression step by step.\n\nFirst, let me write down the expression again to make sure I have it right: $ a(2a + b) - 2a^2 + ab $. Let me expand the first term, which is $ a(2a + b) $. If I distribute the $ a $, that becomes $ 2a^2 + ab $. So substituting that back into the original expression, we have:\n\n$ 2a^2 + ab - 2a^2 + ab $.\n\nNow let me combine like terms. The $ 2a^2 $ and $ -2a^2 $ cancel each other out, right? So that leaves me with $ ab + ab $, which is $ 2ab $. So the entire expression simplifies to $ 2ab $. \n\nWait, so the problem is asking for the number of whole-number divisors of $ 2ab $, where $ a $ and $ b $ are different primes greater than 2. Let me confirm that I simplified correctly. Let me check again:\n\nOriginal expression: $ a(2a + b) - 2a^2 + ab $.\n\nExpanding $ a(2a + b) $: $ 2a^2 + ab $. Then subtract $ 2a^2 $, so $ 2a^2 + ab - 2a^2 = ab $. Then add $ ab $, so $ ab + ab = 2ab $. Yes, that seems right. So the expression simplifies to $ 2ab $.\n\nAlright, so now I need to find the number of positive divisors of $ 2ab $. Since $ a $ and $ b $ are primes greater than 2, they are both odd primes. Let me recall that the number of divisors of a number is found by taking its prime factorization, adding one to each of the exponents, and multiplying those together.\n\nSo first, let me write the prime factorization of $ 2ab $. Since 2 is a prime number, and $ a $ and $ b $ are different primes, all three primes are distinct. Therefore, the prime factors of $ 2ab $ are 2, $ a $, and $ b $. Each of these primes has an exponent of 1 in the factorization. So the prime factorization is $ 2^1 \\times a^1 \\times b^1 $.\n\nTo find the number of divisors, I take each exponent (which are all 1s), add 1 to them, and multiply. That would be $ (1 + 1)(1 + 1)(1 + 1) = 2 \\times 2 \\times 2 = 8 $. Therefore, there are 8 positive divisors.\n\nBut let me verify this with an example to make sure I haven't made a mistake. Let me pick specific primes for $ a $ and $ b $. Let's take $ a = 3 $ and $ b = 5 $, which are both primes greater than 2 and different.\n\nCalculating $ 2ab $: $ 2 \\times 3 \\times 5 = 30 $. Now, the divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. That's 8 divisors. Yep, that matches the calculation. Let me try another example to be safe. Let me take $ a = 5 $, $ b = 7 $. Then $ 2ab = 2 \\times 5 \\times 7 = 70 $. Divisors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Again, 8 divisors. Okay, so that seems consistent.\n\nBut let me think again if there's any possibility that 2 could be equal to $ a $ or $ b $. However, the problem states that $ a $ and $ b $ are primes greater than 2, so they are odd primes. Therefore, 2 is not among them. Therefore, 2, $ a $, and $ b $ are distinct primes. Therefore, the prime factorization is indeed three distinct primes each to the first power, leading to 2×2×2=8 divisors.\n\nBut let me check if there's a case where maybe $ a $ or $ b $ is 2. But the problem says they are greater than 2, so they can't be. Therefore, my reasoning holds.\n\nTherefore, the number of whole-number divisors is 8. Therefore, the answer is 8, which in boxed form is \\boxed{8}.\n\nWait, but let me just check once more. Suppose someone might think that maybe the original expression simplifies differently? Let me check again:\n\nOriginal expression: $ a(2a + b) - 2a^2 + ab $.\n\nFirst term: $ a(2a + b) = 2a^2 + ab $.\n\nSubtracting $ 2a^2 $: $ 2a^2 + ab - 2a^2 = ab $.\n\nAdding $ ab $: $ ab + ab = 2ab $. Yes, that's correct. So the expression simplifies to 2ab. So no issues there.\n\nAlternatively, maybe someone might misread the original problem? Let me check again. The problem says: \"the integer $ a(2a + b) - 2a^2 + ab $\". So yes, that's how it's written. So the steps are correct.\n\nTherefore, I think my answer is correct. The number of divisors is 8.\n\n**Final Answer**\n\\boxed{8}\n</think>\n\nWe are given an expression involving two distinct prime numbers $ a $ and $ b $, both greater than 2, and asked to find the number of whole-number divisors of the integer:\n\n$$\na(2a + b) - 2a^2 + ab\n$$\n\n---\n\n### Step 1: Simplify the Expression\n\nStart by expanding the first term:\n\n$$\na(2a + b) = 2a^2 + ab\n$$\n\nNow substitute this back into the full expression:\n\n$$\n2a^2 + ab - 2a^2 + ab\n$$\n\nSimplify by combining like terms:\n\n- $ 2a^2 - 2a^2 = 0 $\n- $ ab + ab = 2ab $\n\nSo the entire expression simplifies to:\n\n$$\n2ab\n$$\n\n---\n\n### Step 2: Prime Factorization of $ 2ab $\n\nSince $ a $ and $ b $ are distinct prime numbers greater than 2, and 2 is also a prime, the expression $ 2ab $ is the product of three **distinct** primes:\n\n$$\n2ab = 2^1 \\cdot a^1 \\cdot b^1\n$$\n\n---\n\n### Step 3: Count the Number of Divisors\n\nThe number of positive divisors of a number is determined by taking the product of one more than each exponent in its prime factorization.\n\nHere, the exponents are all 1:\n\n$$\n(1 + 1)(1 + 1)(1 + 1) = 2 \\cdot 2 \\cdot 2 = 8\n$$\n\n---\n\n### Final Answer\n\n$$\n\\boxed{8}\n$$", "role": "assistant", "reasoning_content": "" }, "review": { "gold": "8", "pred": "8", "result": true } } ], "created": 1755157209, "model": "Qwen3-32B", "object": "chat.completion", "usage": { "completion_tokens": 1627, "prompt_tokens": 68, "total_tokens": 1695 }, "model_spec": { "api_url": "http://0.0.0.0:9009/v1/chat/completions", "model_id": "Qwen3-32B", "api_key": "EMPTY" }, "answer_id": "answer-1a07ae97260436f698e63d4620d7cff0", "subset_name": "default", "raw_input": { "problem": "Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\cdot a^1\\cdot b^1$ has $(1+1)(1+1)(1+1)=\\boxed{8}$ divisors.", "answer": "8", "subject": "Number Theory", "level": 4, "unique_id": "test/number_theory/1287.json" }, "origin_prompt": { "data": [ "Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?\nPlease reason step by step, and put your final answer within \\boxed{}." ], "system_prompt": null, "raw_input": { "problem": "Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\cdot a^1\\cdot b^1$ has $(1+1)(1+1)(1+1)=\\boxed{8}$ divisors.", "answer": "8", "subject": "Number Theory", "level": 4, "unique_id": "test/number_theory/1287.json" }, "index": 0 }, "reviewed": true, "review_id": "review-307a3dc16716316d92e62e1f3ae09703", "reviewer_spec": { "metric": [ "AveragePass@1" ], "reviewer": [ "Evaluator" ], "revision": [ "default" ] }, "review_time": 1755194244.6042051 }
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", "solution": "Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\\\\cdot a^1\\\\cdot b^1$ ...
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