int main()
{
int T, ca = 1;
scanf("%d", &T);
while(T--) {
scanf("%d%d", &k, &ret);
int m = (int)sqrt(1.0 * MOD) + 10;
std::map<int, int> mp;
for(int i = 0; i < m; i++) {
int tmp = pow_mod(k, i);
if(mp.find(tmp) == mp.end()) {
mp[tmp] = i;
}
}
int ans = MOD;
for(int i = 0; i < m; i++) {
int tmp = pow_mod(pow_mod(k, i), m);
int x, y;
exgcd(tmp, MOD, x, y);
while(x < 0) x += MOD;
x = 1LL * x * ret % MOD;
y = 1LL * y * ret % MOD;
if(mp.find(x) != mp.end()) {
tmp = i * m + mp[x];
if(tmp < ans) {
ans = tmp;
}
}
}
printf("Case %d: %d\n", ca++, ans);
}
return 0;
}
int main()
{
phi_table(maxn);
scanf("%lld %d", &N, &K);
if (K == 1 || K == 3)
{
inv = get_inv(2);
}
else if (K == 2)
{
inv = get_inv(6);
}
else if (K == 4)
{
inv = get_inv(30);
}
else if (K == 5)
{
inv = get_inv(12);
}
long long ans = 0;
for (int i = 1, j = N; i <= j; ++i, j = N / i)
{
printf("%lld %lld\n", pow_mod(i, K), euler_phi(N / i));
ans = (ans + pow_mod(i, K) * euler_phi(N / i) % mod) % mod;
ans = (ans + (sum(j, K) - sum(N / (i + 1), K)) * euler_phi(i) % mod) % mod;
}
if (ans < 0)
{
ans += mod;
}
printf("%lld\n", ans);
return 0;
}
int BSGS(int a, int b, int p) {
int m, v, e = 1, i;
m = (int) sqrt(p + 0.5) + 1;//这里不加1会出错的！
v = pow_mod(pow_mod(a, m, p), p - 2, p);
map<int, int> x;
x[1] = 0;
for (i = 1; i < m; ++i){
e = e * a % p;
if (!x.count(e)) x[e] = i;
}
for (i = 0; i < m; ++i){
if (x.count(b)) return i * m + x[b];
b = b * v % p;
}
return -1; // No solution
}
bool is_primitive(long long a,long long p){
if(a>=p)return 0;
assert(is_prime(p));
vector<pll> d=factorG(p-1);
for(auto i:d) if(pow_mod(a,(p-1)/i.x,p)==1)return 0;
return 1;
}
vector <int> residue(int p,int N,int a)
{
int g=primitive_root(p);
long long m=discrete_log(g,a,p);
vector<int> ret;
if (a==0)
{
ret.push_back(0);
return ret;
}
if (m==-1) return ret;
long long A=N,B=p-1,C=m,x,y;
long long d=extended_gcd(A,B,x,y);
if (c%d!=0) return ret;
x=x*(C/d)%B;
long long delta=B/d;
for (int i=0;i<d;i++)
{
x=((x+delta)%B+B)%B;
ret.push_back((int)pow_mod(g,x,p));
}
sort(ret.begin(),ret.end());
ret.erase(unqinue(ret.begin(),ret.end()),ret.end());
return ret;
}
int main(){
FILE *ff,*fi;
LL ttt,res;
freopen("data.in","r",stdin);
fi = fopen("data.out","r");
ff = fopen("forcedata_0.out","w");
while(scanf("%d%d%d",&n,&m,&k)!=EOF){
memset(g,0,sizeof(g));
//if(n>8000)break;
for(i=0;i<m;i++){
scanf("%d%d%",&ti,&tj);
//printf("%d %d\n",n,tj);
add(ti,tj);
}
result = 0;
for(i=0;i<n;i++)
for(j=0;j<n;j++){
if(!g[i][j]){
result++;
add(i,j);
}
}
//printf("%I64d\n",result);
fscanf(fi,"%I64d",&ttt);
result = pow_mod(k,result,MM);
printf("%I64d %s\n",result,ttt==result?"ok":"!!!!!!!!!!!!!!!!!!!!!!");
if(ttt!=result)scanf(" ");
fprintf(ff,"%I64d\n",result);
}
scanf(" ");
}
int main() {
memset(prime, 0, sizeof(prime));
P();
while (scanf("%d", &n) && n) {
int flag = judge(n);
if (flag) {
printf("%d is normal.\n", n);
continue;
}
else {
int flag1 = 1;
for (int i = 2; i < n; i++) {
int k = pow_mod(i, n, n);
if (k != i) {
flag1 = 0;
break;
}
}
if (flag1)
printf("The number %d is a Carmichael number.\n", n);
else
printf("%d is normal.\n", n);
}
}
return 0;
}
long long pow_mod(int n, int k, int m) {
if(!k) return 1;
long long p = pow_mod(n, k/2, m);
p = p * p % m;
if(k&1) p = p * n % m;
return p;
}
int solve (int d, int M) {
if (d == k - 1)
return A[d]%M;
int phi = euler_phi(M);
int c = solve (d+1, phi) + phi;
return pow_mod(A[d], c, M);
}
int main()
{
int m, n, t;
scanf("%d %d %d", &m, &n, &t);
for (int i = 0; i <= t; ++i)
{
C[i][0] = 1;
for (int j = 1; j < i; ++j)
C[i][j] = (C[i - 1][j - 1] + C[i - 1][j]) % mod;
C[i][i] = 1;
}
int ans = 0;
for (int i = 0; i <= t; ++i)
ans = (ans + pow_mod(m - 2 * i, n) * C[t][i]) % mod;
printf("%d\n", (ans * pow_mod(inv2, t) % mod + mod) % mod);
return 0;
}
LL pow_mod(int a, int n, int m) {
if (n == 1)
return a % m;
LL x = pow_mod(a, n / 2, m);
LL ans = (x * x) % m;
if (n % 2 == 1)
ans = ans * a % m;
return ans;
}
/*
*素性测试，Miller_Rabin 测试算法
*if n < 3,215,031,751, just test a = 2, 3, 5, and 7;
*if n < 4,759,123,141, just test a = 2, 7, and 61;
*if n < 1,122,004,669,633, just test a = 2, 13, 23, and 1662803;
*if n < 2,152,302,898,747, just test a = 2, 3, 5, 7, and 11;
*if n < 3,474,749,660,383, just test a = 2, 3, 5, 7, 11, and 13;
*if n < 341,550,071,728,321, just test a = 2, 3, 5, 7, 11, 13, and 17.
*if n < 3,825,123,056,546,413,051, just test a = 2, 3, 5, 7, 11, 13, 17, 19, and 23.
*if n < 18,446,744,073,709,551,616 = 2^64, just test a = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37.
*/
bool test(ll x, ll n) {
ll m = n - 1;
while (!(m&1)) m >>= 1;
x = pow_mod(x, m, n);
for (; m<n-1 && x!=1 && x!=n-1; m<<=1) {
x = mul_mod(x, x, n); //防爆long long
}
return x == n-1 || (m&1) == 1;
}
// Public library function. Returns NULL if successful, a string starting with "I/O error: "
// if an I/O error occurred (please see perror()), or a string if some other error occurred.
const char *modify_file_crc32(const char *path, uint64_t offset, uint32_t newcrc, bool printstatus) {
FILE *f = fopen(path, "r+b");
if (f == NULL)
return "I/O error: fopen";
// Read entire file and calculate original CRC-32 value.
// Note: We can't use fseek(f, 0, SEEK_END) + ftell(f) to determine the length of the file, due to undefined behavior.
// To be portable, we also avoid using POSIX fseeko()+ftello() or Windows GetFileSizeEx()/_filelength().
uint64_t length;
uint32_t crc = get_crc32_and_length(f, &length);
if (offset > UINT64_MAX - 4 || offset + 4 > length) {
fclose(f);
return "Error: Byte offset plus 4 exceeds file length";
}
if (printstatus)
fprintf(stdout, "Original CRC-32: %08" PRIX32 "\n", reverse_bits(crc));
// Compute the change to make
uint32_t delta = crc ^ newcrc;
delta = (uint32_t)multiply_mod(reciprocal_mod(pow_mod(2, (length - offset) * 8)), delta);
// Patch 4 bytes in the file
fseek64(f, offset);
for (int i = 0; i < 4; i++) {
int b = fgetc(f);
if (b == EOF) {
fclose(f);
return "I/O error: fgetc";
}
b ^= (int)((reverse_bits(delta) >> (i * 8)) & 0xFF);
if (fseek(f, -1, SEEK_CUR) != 0) {
fclose(f);
return "I/O error: fseek";
}
if (fputc(b, f) == EOF) {
fclose(f);
return "I/O error: fputc";
}
if (fflush(f) == EOF) {
fclose(f);
return "I/O error: fflush";
}
}
if (printstatus)
fprintf(stdout, "Computed and wrote patch\n");
// Recheck entire file
bool match = get_crc32_and_length(f, &length) == newcrc;
fclose(f);
if (match) {
if (printstatus)
fprintf(stdout, "New CRC-32 successfully verified\n");
return NULL; // Success
} else
return "Assertion error: Failed to update CRC-32 to desired value";
}
vector<int> multiply(vector<int> u, vector<int> v) { // the same as the complex version
nth_root.clear();
int r = pow_mod(ROOT, (MOD - 1) / n, MOD);
for (int i = 0, tmp = 1; i < n; ++i) {
nth_root.push_back(tmp);
tmp = (long long)tmp * r % MOD;
}
int inv = inverse(n, MOD);
return ret;
}
int main() {
int T, n, k, m, a;
scanf("%d", &T);
for(int i = 1; i <= T; i++) {
scanf("%d", &n);
long long ans = pow_mod(2,n-1,MOD) * n % MOD;
printf("Case #%d: %d\n", i, (int)ans);
}
return 0;
}
//以a为基,n-1=x*2^t a^(n-1)=1(mod n) 验证n是不是合数
//一定是合数返回true,不一定返回false
bool check(long long a,long long n,long long x,long long t){
long long ret=pow_mod(a,x,n);
long long last=ret;
for(int i=1;i<=t;i++){
ret=mult_mod(ret,ret,n);
if(ret==1&&last!=1&&last!=n-1) return true;//合数
last=ret;
}
if(ret!=1) return true;
return false;
}
int main()
{
int t,n;
while(scanf("%d",&t)!=EOF)
while(t--)
{
scanf("%d",&n);
printf("%lld\n",pow_mod(2,n-1,mod)-1);
}
return 0;
}
void fermat_test(const int n, int* p, int* np) {
*p = *np = 0;
int a;
#pragma omp parallel for private(a)
for ( a=2; a < n; a++) {
if ( pow_mod(a, n, n) == a)
*p += 1;
else
*np +=1;
}
}
bool test(int n, int a, int d) {
if (n == 2) return true;
if (n == a) return true;
if ((n & 1) == 0) return false;
while (!(d & 1)) d = d >> 1;
int t = pow_mod(a, d, n);
while ((d != n - 1) && (t != 1) && (t != n - 1)) {
t = (long long)t * t % n;
d = d << 1;
}
return (t == n - 1 || (d & 1) == 1);
}
// Solve the congruence x^2 = n (mod p).
void tonelli_shanks(uint32_t n, uint32_t p, uint32_t *result) {
if(p == 2) {
result[0] = n;
result[1] = n;
return;
}
uint64_t S = 0;
uint64_t Q = p - 1;
while(Q % 2 == 0) {
Q /= 2;
++S;
}
uint64_t z = 2;
while(legendre_symbol(z, p) != -1)
++z;
uint64_t c = pow_mod(z, Q, p);
uint64_t R = pow_mod(n, (Q + 1) / 2, p);
uint64_t t = pow_mod(n, Q, p);
uint64_t M = S;
while(t % p != 1) {
int32_t i = 1;
while(pow_mod(t, pow(2, i), p) != 1)
++i;
uint64_t b = pow_mod(c, pow(2, M - i - 1), p);
R = R * b % p;
t = t * b * b % p;
c = b * b % p;
M = i;
}
result[0] = R;
result[1] = p - R;
}
int modsqr(int a,int b=n)
{
int b,k,i,x;
if(n==2)return a%n;
if(pow_mod(a,(n-1)/2,n)==1)
{
if(n%4==3)
{
x=pow_mod(a,(n+1)/4,n);
}
else
{
for(b=1;pow_mod(b,(n-1)/2,n)==1;b++);
i=(n-1)/2;
k=0;
do
{
i/=2;k/=2;
if((pow_mod(a,i,n)*(long long)pow(b,k,n)+1)%n==0)
k+=(n-1)/2;
}while(i%2=0);
x=(pow_mod(a,(i+1)/2,n)*(long long)pow_mod(b,k/2,n))%n;
}
if(x*2>n)
{
x=n-x;
}
return x;
}
return -1;
}
int main()
{
scanf("%s", s);
scanf("%s", t);
int n = strlen(s);
int k;
scanf("%d", &k);
long long cnt = (pow_mod(n - 1, k) + (k & 1 ? 1 : -1)) * pow_mod(n, mod - 2) % mod;
long long cnt0 = cnt - (k & 1 ? 1 : -1);
long long ans = 0;
for (int i = 0; i < n; ++i)
{
bool flag = true;
for (int j = 0; j < n; ++j)
if (s[j] != t[(j + i) % n])
flag = false;
if (flag)
ans = (ans + (i == 0 ? cnt0 : cnt)) % mod;
}
printf("%I64d\n", (ans + mod) % mod);
return 0;
}
int rsa_sign(unsigned char** result, unsigned char* hash, unsigned int hashLengthBytes, unsigned char* privateKey, unsigned char* modulus, unsigned int keyLengthBytes){
if(hashLengthBytes>(keyLengthBytes-11)){
return 0;
}
unsigned char* padded=new unsigned char[keyLengthBytes];
add_PKCS1_padding(padded,hash,hashLengthBytes,false,keyLengthBytes);
int r=pow_mod(result,padded,privateKey,modulus,keyLengthBytes);
return r;
}
bool solve(int n)
{
for (int i = 0; i <= 23; ++i)
{
int s = 1 << i;
if (i != 23 && pow_mod(n, s * 3 * 5 * 7) == 1) return false;
if (pow_mod(n, s * 3 * 5 * 1) == 1) return false;
if (pow_mod(n, s * 3 * 1 * 7) == 1) return false;
if (pow_mod(n, s * 3 * 1 * 1) == 1) return false;
if (pow_mod(n, s * 1 * 5 * 7) == 1) return false;
if (pow_mod(n, s * 1 * 5 * 1) == 1) return false;
if (pow_mod(n, s * 1 * 1 * 7) == 1) return false;
if (pow_mod(n, s * 1 * 1 * 1) == 1) return false;
}
return true;
}
inline int calc(int a, int p) {
if (a < 2)
return a;
if (!quad_resi(a, p))
return p; // no solution
if (p % 4 == 3)
return pow_mod(a, (p + 1) / 4, p);
int b = 0;
while (quad_resi((my_sqr(b) - a + p) % p, p))
b = rand() % p;
quad_poly ret = quad_poly(b, 1, (my_sqr(b) - a + p) % p, p);
ret = ret.pow((p + 1) / 2);
return ret.zero;
}
bool witness(LL a, LL n, LL x, LL t) //以a为基，n-1=x*2^t，检验n是不是合数
{
LL ret = pow_mod(a, x, n);
LL last = ret;
for (int i = 1; i <= t; i++)
{
ret = muti_mod(ret, ret, n);
if (ret == 1 && last != 1 && last != n - 1)
return 1;
last = ret;
}
if (ret != 1) return 1;
return 0;
}
bool test(long long n, long long a, long long d)
{
if (n == 2) return true;
if (n == a) return true;
if ((n & 1) == 0) return false;
while(!(d & 1)) d = d >> 1;
long long t = pow_mod(a, d, n);
while((d != n - 1) && (t != 1) && (t != n - 1))
{
t = t * t % n;
d = d << 1;
}
return (t == n - 1 || (d & 1) == 1);
}
bool check(LL a, LL n, LL x, LL t)
{ //以a为基，n-1=x*2^t，检验n是不是合数
LL ret = pow_mod(a, x, n), last = ret;
for (int i = 1; i <= t; i++)
{
ret = multi_mod(ret, ret, n);
if (ret == 1 && last != 1 && last != n - 1)
return 1;
last = ret;
}
if (ret != 1)
return 1;
return 0;
}
bool Wintess(int64 aa, int64 n, int64 mod) /*是以aa为底的伪质数返回0,是合数返回1*/
{
int t = 0;
while ((n & 1) == 0){ /*n = d * 2^t; n = d;*/
n >>= 1;
++ t;
}
int64 x = pow_mod(aa, n, mod), y;
while (t --){
y = mul_mod (x, x, mod);
if (y == 1 && x != 1 && x != mod-1) return 1;
x = y;
}
return (y != 1);
}
// get a^p mod n
int pow_mod(int a, int n, int p){
if (p == 1) {
return a;
}
int half_p = p/2;
int q = p%2;
int t;
//#pragma omp task shared(t)
t = pow_mod(a, n, half_p);
int r = (t*t) %n;
//#pragma omp taskwait
if (q==0)
return r;
else
return ((r*a)%n);
}
