UVa 408 Uniform generator: gcd & Proof

408-uniform Generator

Time limit:3.000 seconds

Http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=100&page=show_ problem&problem=349

Computer simulations often require random numbers. One way to generate pseudo-random numbers are via a function of the form

Where ' is the modulus operator.

Such a function would generate pseudo-random numbers (seed) between 0 and MOD-1. One problem with functions of the ' this ' form is ' that they'll always generate the ' same '. In order to minimize this effect, selecting the "step" and MOD values carefully can result in a uniform distribution of all Values between (and including) 0 and MOD-1.

For example, if step = 3 and MOD = 5, the function would generate the series of pseudo-random numbers 0, 3, 1, 4, 2 in a re Peating cycle. In this example, all of the numbers between and including 0 and MOD-1 would be generated every MOD iterations of the Functi On. Note "By" The nature of the "function to generate" sameseed (x+1) every time seed (x) occurs means that if a function w Ill generate all the numbers between 0 andMOD-1, it'll generate pseudo-random the numbers with uniformly MOD every S.

If step = The function generates the series 0, 5 (or any of the other repeating series if the initial SE Ed is other than 0). This is a poor selection of step and MOD because no initial seed would generate all of the numbers from 0 and MOD-1.

Your program would determine if choices of step and MOD'll generate a uniform distribution of pseudo-random numbers.

Input

Each line of input would contain a pair of integers for step and MOD in-order ( ).

Output

For each line of input, your program should print the step value right-justified in columns 1 through, the MOD value R Ight-justified in columns through and either ' good Choice ' or ' bad Choice ' left-justified starting in column 25. The ' good Choice ' message should is printed the selection of the step andmod'll generate all the numbers between and I Ncluding 0 and MOD-1 when MOD numbers are generated. Otherwise, your should print the message ' Bad Choice '. After each output test set, the your program should print exactly one blank line.

Sample Input

3 5
63923 99999

Sample Output

3 5 Good Choice

Bad Choice

63923 99999 Good Choice

The first can be directly calculated by brute force:

/*0.082s*/
    
#include <cstdio>  
    
int main ()  
{  
    int step, mod, X, Count;  
    while (~SCANF ("%d%d", &step, &mod))  
    {  
        x = 0, count = 0;  
        Do
        {  
            x = (x + step)% MoD;  
            ++count;  
        }  
        while (x && count!= MoD);  
        printf ("%10d%10d    %s\n\n", step, mod, count = mod && x = 0? "Good Choice": "Bad Choice");  
    }  
    return 0;  
}

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Using the above code, you can find that when mod=20, only 1,3,7,9,11,13,17,19,... is a good choice, and these numbers are 20 coprime.

So we guess: when gcd (Step,mod)!=1, this is not a good choice.

Proof: Set GCD (Step,mod) =k, then step=ka,mod=kb, no matter how the calculation

Seed (x+1) ≡seed (x) (mod k)

So there must be some numbers that can't be generated.

Optimized code:

/*0.022s*/
    
#include <cstdio>  
    
int gcd (int a, int b)  
{return  
    b gcd (b, a% B): A;  
}  
    
int main ()  
{  
    int step, mod;  
    while (~SCANF ("%d%d", &step, &mod)  
        printf ("%10d%10d    %s\n\n", step, mod, gcd (step, MoD) = 1? "Good Choice": "Bad Choice");  
    return 0;  
}