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E - LCM on Whiteboard

Score : $500$ points

Problem Statement

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:

$N$

$m_1$

$p_{1,1}$ $e_{1,1}$

$\vdots$

$p_{1,m_1}$ $e_{1,m_1}$

$m_2$

$p_{2,1}$ $e_{2,1}$

$\vdots$

$p_{2,m_2}$ $e_{2,m_2}$

$\vdots$

$m_N$

$p_{N,1}$ $e_{N,1}$

$\vdots$

$p_{N,m_N}$ $e_{N,m_N}$

Output

Print the answer.

4
1
7 2
2
2 2
5 1
1
5 1
2
2 1
7 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 $3$.

1
1
998244353 1000000000

1

There may be enormous integers on the whiteboard.