POJ3696 The Luckiest number

题意

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The Luckiest number

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 7083		Accepted: 1886

Description

Chinese people think of '8' as the lucky digit. Bob also likes digit '8'. Moreover, Bob has his own lucky number L. Now he wants to construct his luckiest number which is the minimum among all positive integers that are a multiple of L and consist of only digit '8'.

Input

The input consists of multiple test cases. Each test case contains exactly one line containing L(1 ≤ L ≤ 2,000,000,000).

The last test case is followed by a line containing a zero.

Output

For each test case, print a line containing the test case number( beginning with 1) followed by a integer which is the length of Bob's luckiest number. If Bob can't construct his luckiest number, print a zero.

Sample Input

8
11
16
0

Sample Output

Case 1: 1
Case 2: 2
Case 3: 0

Source

2008 Asia Hefei Regional Contest Online by USTC

分析

\[\because L | \frac{8(10^x-1)}{9} \\
\therefore 9L | 8(10^x-1) \\
\therefore \frac{9L}{\gcd(L,8)} | 10^x-1 \\
\therefore 10^x \equiv 1 (\bmod \frac{9L}{\gcd(L,8)}) \\
\because a^x \equiv 1 (\bmod n) \rightarrow \gcd(a,n)=1,x|\varphi(n) \\
\therefore x|\varphi(\frac{9L}{\gcd(L,8)})
\]

所以枚举即可。时间复杂度\(O(\sqrt{L} \log L)\)

代码

#include<iostream>
#include<cmath>
#define rg register
#define il inline
#define co const
template<class T>il T read(){
	rg T data=0,w=1;
	rg char ch=getchar();
	while(!isdigit(ch)){
		if(ch=='-') w=-1;
		ch=getchar();
	}
	while(isdigit(ch))
		data=data*10+ch-'0',ch=getchar();
	return data*w;
}
template<class T>il T read(rg T&x){
	return x=read<T>();
}
typedef long long ll;

ll L,d,k,p,s,i,num=0;
ll gcd(ll x,ll y){
	return y?gcd(y,x%y):x;
}
ll ksc(ll a,ll b,ll c){
	ll ans=0;
	while(b){
		if(b&1) ans=(ans+a)%c;
		a=a*2%c;
		b>>=1;
	}
	return ans;
}
ll ksm(ll a,ll b,ll c){
	ll ans=1%c;
	a%=c;
	while(b){
		if(b&1) ans=ksc(ans,a,c);
		a=ksc(a,a,c);
		b>>=1;
	}
	return ans;
}
ll phi(ll n){
	ll ans=n;
	for(i=2;i*i<=n;++i)
		if(n%i==0){
			ans=ans/i*(i-1);
			while(n%i==0) n/=i;
		}
	if(n>1) ans=ans/n*(n-1);
	return ans;
}
ll number(){
	d=gcd(L,8);
	k=9*L/d;
	if(gcd(k,10)!=1) return 0;
	p=phi(k);
	s=sqrt((double)p);
	for(i=1;i<=s;++i)
		if(p%i==0&&ksm(10,i,k)==1)
			return i;
	for(i=s-1;i;--i)
		if(p%i==0&&ksm(10,p/i,k)==1)
			return p/i;
	return 0;
}
int main(){
//	freopen(".in","r",stdin);
//	freopen(".out","w",stdout);
	while(read(L)) printf("Case %lld: %lld\n",++num,number());
	return 0;
}