Problem p02722 (C++)

Buggy submission (status: Time Limit Exceeded):
#include <algorithm>
#include <bitset>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <vector>

using namespace std;

#define DEBUG_ //!!提出時にコメントアウト!!
#ifdef DEBUG_
#define dump(x) cerr << #x << " = " << (x) << endl;
#else
#define dump(x) ;
#endif

#define equals(a, b) (fabs((a) - (b)) < EPS)
#define FOR(i, a, b) for (int i = (a); i < (b); ++i)
#define REP(i, n) FOR(i, 0, n)
#define SZ(x) ((int)(x).size())
#define pb push_back
#define eb emplace_back

// #define int long long

typedef long long LL;
typedef vector<int> VI;
typedef vector<LL> VL;
typedef vector<VI> VVI;
typedef vector<string> VS;
typedef pair<int, int> PII;
typedef pair<LL, LL> PLL;

template <class T> bool chmax(T &a, const T &b) {
  if (a < b) {
    a = b;
    return 1;
  }
  return 0;
}
template <class T> bool chmin(T &a, const T &b) {
  if (b < a) {
    a = b;
    return 1;
  }
  return 0;
}

template <typename T> std::string printVector(const std::vector<T> &data) {
  std::stringstream ss;
  std::ostream_iterator<T> out_it(ss, ", ");
  ss << "[";
  std::copy(data.begin(), data.end() - 1, out_it);
  ss << data.back() << "]";
  return ss.str();
}

template <typename T> void print_array(const T &ary, int size) {
  REP(i, size) { cout << ary[i] << " "; }
  cout << endl;
}

const int MOD = 1e9 + 7;
const LL LINF = 1001002003004005006ll;
const int INF = 1001001001;
const double EPS = (1e-10);
LL N;

bool valid(LL x) {
  LL A = N;
  while (A >= x) {
    if (A % x == 0) {
      A /= x;
    } else {
      A %= x;
    }
  }
  if (A == 1)
    return true;
  else
    return false;
}

signed main(int argc, char const *argv[]) {
  cin.tie(0);
  ios::sync_with_stdio(false);
  cout << setprecision(12);

  cin >> N;
  set<LL> cand;
  for (LL i = 2; i * i <= N - 1; i++) {
    if ((N - 1) % i == 0) {
      cand.insert(i);
      cand.insert((N - 1) / i);
    }
  }
  for (LL i = 2; i * i <= N; i++) {
    if ((N) % i == 0) {
      cand.insert(i);
      cand.insert((N) / i);
    }
  }
  cand.insert(N);
  cand.insert(N - 1);
  LL ans = 0;
  for (LL i : cand) {
    if (valid(i)) {
      ans++;
    }
  }
  cout << ans << endl;
}


Fixed submission:
#include <algorithm>
#include <bitset>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <vector>

using namespace std;

#define DEBUG_ //!!提出時にコメントアウト!!
#ifdef DEBUG_
#define dump(x) cerr << #x << " = " << (x) << endl;
#else
#define dump(x) ;
#endif

#define equals(a, b) (fabs((a) - (b)) < EPS)
#define FOR(i, a, b) for (int i = (a); i < (b); ++i)
#define REP(i, n) FOR(i, 0, n)
#define SZ(x) ((int)(x).size())
#define pb push_back
#define eb emplace_back

// #define int long long

typedef long long LL;
typedef vector<int> VI;
typedef vector<LL> VL;
typedef vector<VI> VVI;
typedef vector<string> VS;
typedef pair<int, int> PII;
typedef pair<LL, LL> PLL;

template <class T> bool chmax(T &a, const T &b) {
  if (a < b) {
    a = b;
    return 1;
  }
  return 0;
}
template <class T> bool chmin(T &a, const T &b) {
  if (b < a) {
    a = b;
    return 1;
  }
  return 0;
}

template <typename T> std::string printVector(const std::vector<T> &data) {
  std::stringstream ss;
  std::ostream_iterator<T> out_it(ss, ", ");
  ss << "[";
  std::copy(data.begin(), data.end() - 1, out_it);
  ss << data.back() << "]";
  return ss.str();
}

template <typename T> void print_array(const T &ary, int size) {
  REP(i, size) { cout << ary[i] << " "; }
  cout << endl;
}

const int MOD = 1e9 + 7;
const LL LINF = 1001002003004005006ll;
const int INF = 1001001001;
const double EPS = (1e-10);
LL N;

bool valid(LL x) {
  LL A = N;
  while (A >= x) {
    if (A % x == 0) {
      A /= x;
    } else {
      A %= x;
    }
  }
  if (A == 1)
    return true;
  else
    return false;
}

signed main(int argc, char const *argv[]) {
  cin.tie(0);
  ios::sync_with_stdio(false);
  cout << setprecision(12);

  cin >> N;
  set<LL> cand;
  for (LL i = 2; i * i <= N - 1; i++) {
    if ((N - 1) % i == 0) {
      cand.insert(i);
      cand.insert((N - 1) / i);
    }
  }
  for (LL i = 2; i * i <= N; i++) {
    if ((N) % i == 0) {
      cand.insert(i);
      cand.insert((N) / i);
    }
  }
  cand.insert(N);
  if (N != 2)
    cand.insert(N - 1);
  LL ans = 0;
  for (LL i : cand) {
    if (valid(i)) {
      ans++;
    }
  }
  cout << ans << endl;
}