A^n - 1

AtCoder

IDarc191_c

Time2000ms

Memory256MB

Difficulty—

You are given a positive integer $N$ between $1$ and $10^9$, inclusive. Find one pair of positive integers $(A, M)$ satisfying the following conditions. It can be proved that such a pair of integers always exists under the constraints. * Both $A$ and $M$ are positive integers between $1$ and $10^{18}$, inclusive. * There exists a positive integer $n$ such that $A^n - 1$ is a multiple of $M$, and the smallest such $n$ is $N$. You are given $T$ test cases; solve each of them. ## Constraints * $1 \le T \le 10^4$ * $1 \le N \le 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Here, $\text{case}_i$ denotes the $i$\-th test case. Each test case is given in the following format: $N$ [samples]

Samples

Input #1

Output #1

2 7
11 68
20250126 1
33 662

Consider $\text{case}_1$.
For example, if we choose $(A,M)=(2,7)$, then:

*   When $n=1$: $2^1 - 1 = 1$ is not a multiple of $7$.
*   When $n=2$: $2^2 - 1 = 3$ is not a multiple of $7$.
*   When $n=3$: $2^3 - 1 = 7$ is a multiple of $7$.

Hence, the smallest $n$ for which $A^n - 1$ is a multiple of $M$ is $3$. Therefore, $(A,M)=(2,7)$ is a correct solution. Other valid solutions include $(A,M)=(100,777)$.

API Response (JSON)

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