LCM on Whiteboard

AtCoder

IDabc259_e

Time2000ms

Memory256MB

Difficulty—

There are $N$ integers $a_1,\ldots,a_N$ written on a whiteboard. Here, $a_i$ can be represented as $a_i = p_{i,1}^{e_{i,1}} \times \ldots \times p_{i,m_i}^{e_{i,m_i}}$ using $m_i$ prime numbers $p_{i,1} \lt \ldots \lt p_{i,m_i}$ and positive integers $e_{i,1},\ldots,e_{i,m_i}$. You will choose one of the $N$ integers to replace it with $1$. Find the number of values that can be the least common multiple of the $N$ integers after the replacement. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq m_i$ * $\sum{m_i} \leq 2 \times 10^5$ * $2 \leq p_{i,1} \lt \ldots \lt p_{i,m_i} \leq 10^9$ * $p_{i,j}$ is prime. * $1 \leq e_{i,j} \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $m_1$ $p_{1,1}$ $e_{1,1}$ $\vdots$ $p_{1,m_1}$ $e_{1,m_1}$ $m_2$ $p_{2,1}$ $e_{2,1}$ $\vdots$ $p_{2,m_2}$ $e_{2,m_2}$ $\vdots$ $m_N$ $p_{N,1}$ $e_{N,1}$ $\vdots$ $p_{N,m_N}$ $e_{N,m_N}$ [samples]

Samples

Input #1

Output #1

3

The integers on the whiteboard are $a_1 =7^2=49, a_2=2^2 \times 5^1 = 20, a_3 = 5^1 = 5, a_4=2^1 \times 7^1 = 14$.  
If you replace $a_1$ with $1$, the integers on the whiteboard become $1,20,5,14$, whose least common multiple is $140$.  
If you replace $a_2$ with $1$, the integers on the whiteboard become $49,1,5,14$, whose least common multiple is $490$.  
If you replace $a_3$ with $1$, the integers on the whiteboard become $49,20,1,14$, whose least common multiple is $980$.  
If you replace $a_4$ with $1$, the integers on the whiteboard become $49,20,5,1$, whose least common multiple is $980$.  
Therefore, the least common multiple of the $N$ integers after the replacement can be $140$, $490$, or $980$, so the answer is $3$.

Input #2

1
1
998244353 1000000000

Output #2

1

There may be enormous integers on the whiteboard.

API Response (JSON)

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  "problem": {
    "name": "LCM on Whiteboard",
    "description": {
      "content": "There are $N$ integers $a_1,\\ldots,a_N$ written on a whiteboard.   Here, $a_i$ can be represented as $a_i = p_{i,1}^{e_{i,1}} \\times \\ldots \\times p_{i,m_i}^{e_{i,m_i}}$ using $m_i$ prime numbers $p_{",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
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    "is_sync": true,
    "sync_url": null,
    "sign": "abc259_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ integers $a_1,\\ldots,a_N$ written on a whiteboard.  \nHere, $a_i$ can be represented as $a_i = p_{i,1}^{e_{i,1}} \\times \\ldots \\times p_{i,m_i}^{e_{i,m_i}}$ using $m_i$ prime numbers $p_{...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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