Sequence in mod P

AtCoder

IDabc270_g

Time4000ms

Memory256MB

Difficulty—

There is a sequence $X=(X_0, X_1, \ldots)$ defined by the following recurrence relation. $X_i = \left{ \begin{array}{ll} S & (i = 0)\\ (A X_{i-1}+B) \bmod P & (i \geq 1) \end{array} \right.$ Determine whether there is an $i$ such that $X_i=G$. If it exists, find the smallest such $i$. Here, $x \bmod y$ denotes the remainder when $x$ is divided by $y$ (the least non-negative residue). You are given $T$ test cases for each input file. ## Constraints * $1 \leq T \leq 100$ * $2 \leq P \leq 10^9$ * $P$ is a prime. * $0\leq A,B,S,G < P$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$ Each test case is in the following format: $P$ $A$ $B$ $S$ $G$ [samples]

Samples

Input #1

3
5 2 1 1 0
5 2 2 3 0
11 1 1 0 10

Output #1

3
-1
10

For the first test case, we have $X=(1,3,2,0,\ldots)$, so the smallest $i$ such that $X_i=0$ is $3$.  
For the second test case, we have $X=(3,3,3,3,\ldots)$, so there is no $i$ such that $X_i=0$.

API Response (JSON)

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      "content": "There is a sequence $X=(X_0, X_1, \\ldots)$ defined by the following recurrence relation. $X_i = \\left{ \\begin{array}{ll} S & (i = 0)\\\\ (A X_{i-1}+B) \\bmod P & (i \\geq 1) \\end{array} \\right.$ Determine",
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      "statement_type": "Markdown",
      "content": "There is a sequence $X=(X_0, X_1, \\ldots)$ defined by the following recurrence relation.\n$X_i = \\left{ \\begin{array}{ll} S & (i = 0)\\\\ (A X_{i-1}+B) \\bmod P & (i \\geq 1) \\end{array} \\right.$\nDetermine...",
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