Primitive Primes
You cannot submit for this problem because the contest is ended. You can click "Open in Problem Set" to view this problem in normal mode.
Description
It is Professor R's last class of his teaching career. Every time Professor R taught a class, he gave a special problem for the students to solve. You being his favourite student, put your heart into solving it one last time.
You are given two polynomials $f(x) = a_0 + a_1x + \dots + a_{n-1}x^{n-1}$ and $g(x) = b_0 + b_1x + \dots + b_{m-1}x^{m-1}$, with positive integral coefficients. It is guaranteed that the cumulative GCD of the coefficients is equal to $1$ for both the given polynomials. In other words, $gcd(a_0, a_1, \dots, a_{n-1}) = gcd(b_0, b_1, \dots, b_{m-1}) = 1$. Let $h(x) = f(x)\cdot g(x)$. Suppose that $h(x) = c_0 + c_1x + \dots + c_{n+m-2}x^{n+m-2}$.
You are also given a prime number $p$. Professor R challenges you to find any $t$ such that $c_t$ isn't divisible by $p$. He guarantees you that under these conditions such $t$ always exists. If there are several such $t$, output any of them.
As the input is quite large, please use fast input reading methods.
The first line of the input contains three integers, $n$, $m$ and $p$ ($1 \leq n, m \leq 10^6, 2 \leq p \leq 10^9$), — $n$ and $m$ are the number of terms in $f(x)$ and $g(x)$ respectively (one more than the degrees of the respective polynomials) and $p$ is the given prime number.
It is guaranteed that $p$ is prime.
The second line contains $n$ integers $a_0, a_1, \dots, a_{n-1}$ ($1 \leq a_{i} \leq 10^{9}$) — $a_i$ is the coefficient of $x^{i}$ in $f(x)$.
The third line contains $m$ integers $b_0, b_1, \dots, b_{m-1}$ ($1 \leq b_{i} \leq 10^{9}$) — $b_i$ is the coefficient of $x^{i}$ in $g(x)$.
Print a single integer $t$ ($0\le t \le n+m-2$) — the appropriate power of $x$ in $h(x)$ whose coefficient isn't divisible by the given prime $p$. If there are multiple powers of $x$ that satisfy the condition, print any.
Input
The first line of the input contains three integers, $n$, $m$ and $p$ ($1 \leq n, m \leq 10^6, 2 \leq p \leq 10^9$), — $n$ and $m$ are the number of terms in $f(x)$ and $g(x)$ respectively (one more than the degrees of the respective polynomials) and $p$ is the given prime number.
It is guaranteed that $p$ is prime.
The second line contains $n$ integers $a_0, a_1, \dots, a_{n-1}$ ($1 \leq a_{i} \leq 10^{9}$) — $a_i$ is the coefficient of $x^{i}$ in $f(x)$.
The third line contains $m$ integers $b_0, b_1, \dots, b_{m-1}$ ($1 \leq b_{i} \leq 10^{9}$) — $b_i$ is the coefficient of $x^{i}$ in $g(x)$.
Output
Print a single integer $t$ ($0\le t \le n+m-2$) — the appropriate power of $x$ in $h(x)$ whose coefficient isn't divisible by the given prime $p$. If there are multiple powers of $x$ that satisfy the condition, print any.
3 2 2
1 1 2
2 1
2 2 999999937
2 1
3 1
1
2
Note
In the first test case, $f(x)$ is $2x^2 + x + 1$ and $g(x)$ is $x + 2$, their product $h(x)$ being $2x^3 + 5x^2 + 3x + 2$, so the answer can be 1 or 2 as both 3 and 5 aren't divisible by 2.
In the second test case, $f(x)$ is $x + 2$ and $g(x)$ is $x + 3$, their product $h(x)$ being $x^2 + 5x + 6$, so the answer can be any of the powers as no coefficient is divisible by the given prime.
20240507集训
- Status
- Done
- Rule
- IOI
- Problem
- 6
- Start at
- 2024-5-7 19:00
- End at
- 2024-5-7 21:00
- Duration
- 2 hour(s)
- Host
-
SamuelXuch
- Partic.
- 12