hdoj-1395-2^x mod n = 1【欧拉定理】

2^x mod n = 1

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)

Total Submission(s): 14349 Accepted Submission(s): 4438



[align=left]Problem Description[/align]
Give a number n, find the minimum x(x>0) that satisfies 2^x mod n = 1.

[align=left]Input[/align]
One positive integer on each line, the value of n.

[align=left]Output[/align]
If the minimum x exists, print a line with 2^x mod n = 1.

Print 2^? mod n = 1 otherwise.

You should replace x and n with specific numbers.

[align=left]Sample Input[/align]

2
5

[align=left]Sample Output[/align]

2^? mod 2 = 1
2^4 mod 5 = 1

[align=left]Author[/align]
MA, Xiao

[align=left]Source[/align]
ZOJ Monthly, February 2003

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#include<stdio.h>
int main(){
int n;
while(~scanf("%d",&n)){
if(n==1||n%2==0) printf("2^? mod %d = 1\n",n);
else{
int k=2,ncas=1;
while(1){
if(k%n==1) break;
k=k%n*2;
++ncas;
}
printf("2^%d mod %d = 1\n",ncas,n);
}
}
return 0;
}

欧拉定理 ：
对于互质的正整数 a 和 n ，有 a ^φ(n) ≡ 1 mod n 。 φ(n) 指小于n且与n互质的的正整数（包括1）的个数；
因为2与所有奇数都互质，所以对与奇数n，总会存在x，使得: 2^x =1（mod n）