Kattis

# Discrete Logging

Given a prime $P$, $2 \le P < 2^{31}$, an integer $B$, $2 \le B < P$, and an integer $N$, $1 \le N < P$, compute the discrete logarithm of $N$, base $B$, modulo $P$. That is, find an integer $L \ge 0$ such that

$B^ L \equiv N \bmod {P}.$

The solution to this problem requires a well known result in number theory that is probably expected of you for Putnam but not ACM competitions. It is Fermat’s theorem that states

$B^{P-1} \equiv 1 \bmod {P}$

for any prime $P$ and some other (fairly rare) numbers known as base-$B$ pseudoprimes. A rarer subset of the base-$B$ pseudoprimes, known as Carmichael numbers, are pseudoprimes for every base between $2$ and $P-1$. A corollary to Fermat’s theorem is that for any $m$

$B^{-m} \equiv B^{P-1-m} \bmod {P}$

## Input

There are several lines of input (at most $15$), each containing $P$, $B$, and $N$ separated by a space.

## Output

For each input line, print the logarithm on a separate line. If there are several solutions, print the smallest; if there is none, print “no solution”.

Sample Input 1 Sample Output 1
5 2 1
5 2 2
5 2 3
5 2 4
5 3 1
5 3 2
5 3 3
5 3 4
5 4 1
5 4 2
5 4 3
5 4 4
12345701 2 1111111
1111111121 65537 1111111111

0
1
3
2
0
3
1
2
0
no solution
no solution
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