ll T, N, M, X, Y; ll A[1d6], B[]; ll baka1(){ ll i, j, res = 0; rep(i,N) A[i] = (X*i + Y) % M; rep(i,N) rep(j,i+1,N) if(A[i] > A[j]) res++; return res; } ll baka2(){ ll i, j, res = 0; rep(i,N) B[i] = X*i + Y; rep(i,N) rep(j,i+1,N) res += fDiv(B[i] + M - 1 - (B[j] % M), M) - fDiv(B[i], M); return res; } ll baka3(){ ll i, j, res = 0; rep(i,N) B[i] = X*i + Y; rep(i,N) rep(j,i+1,N) res += fDiv(B[i] + M - 1 - B[j], M) - fDiv(B[i], M) + fDiv(B[j], M); return res; } ll baka4(){ ll i, j, res = 0; rep(i,N) B[i] = X*i + Y; rep(i,N) res += fDiv(B[i], M) * (2 * i + 1 - N); rep(i,N) rep(j,i+1,N) res += fDiv(X * (i-j) + M - 1, M); return res; } ll baka5(){ ll i, res = 0; rep(i,N) res += fDiv(X * i + Y, M) * (2 * i + 1 - N); rep(i,N) res += fDiv(-X * i + M - 1, M) * (N - i); return res; } // ordinary_floor_sum(ll n, ll m, ll a, ll b) // = sum[k=0..n-1] floor( (a*k+b) / m ) // = num of lattice points [0 <= x < n] && [0 < y <= (Ax+B)/M] // modified_floor_sum(ll n, ll m, ll a, ll b, ll x, ll y, ll c) // = sum of lattice points [0 < x <= n] && [0 < y <= (Ax+B)/M] : weight of (X,Y) = x*X + y*Y + c // = sum[X=1..n] [let Y=floor((a*x+b)/m)] (x1 * x + x0) ((y+1) * y1 + 2 * y0) y / 2 // modified_floor_sum(n,m,a,b-a,0,0,1) = ordinary_floor_sum(n,m,a,b) // ordinary_floor_sum: n >= 0, m != 0 // modified_floor_sum: n >= 0, m != 0, other constraints? ll baka_modified_floor_sum_bakabaka(ll n, ll m, ll a, ll b, ll x, ll y, ll c){ ll res = 0, X, Y, h; rep(X,1,n+1){ h = fDiv(a*X+b, m); rep(Y,1,h+1){ res += X*x+Y*y+c; } } return res; } ll baka_modified_floor_sum(ll n, ll m, ll a, ll b, ll x, ll y, ll c){ ll res = 0, X, Y; rep(X,1,n+1){ Y = fDiv(a*X+b, m); res += Y * (Y+1) / 2 * y + Y * (c + x*X); } return res; } ll modified_floor_sum(ll n, ll m, ll a, ll b, ll x, ll y, ll c){ ll res = 0; ll h, d, w, aa, bb; if(a < 0){ return modified_floor_sum(n, m, -a, b + (n+1)*a, -x, y, c+x*(n+1)); } h = fDiv(b, m); if(h){ res += (h * n * ( (h+1)*y + (n+1)*x )) / 2 + h * n * c; b -= m * h; c += y * h; } d = fDiv(a, m); res += n * (n+1) / 2 * d * ( 6*c + y*(2*d*n+d+3) + (4*n+2) * x ) / 6; a -= d * m; x += d * y; if(a == 0) return res; w = fDiv(a*n+b, m); aa = -m; bb = b + a * (n+1); res += modified_floor_sum(w, a, aa, bb, y, -x, c+x*(n+1)); return res; } ll baka6(){ ll i, res = 0; rep(i,1,N+1) res += fDiv(X * i + Y - X, M) * (2 * i - 1 - N); rep(i,1,N+1) res += fDiv(-X * i + M - 1 + X, M) * (N - i + 1); return res; } ll baka7(){ ll i, res = 0; res += modified_floor_sum(N, M, X, Y-X, 2, 0, -1-N); res += modified_floor_sum(N, M, -X, M-1+X, -1, 0, N+1); return res; } ll solve(){ ll i, res = 0; res += modified_floor_sum(N, M, X, Y-X, 2, 0, -1-N); res += modified_floor_sum(N, M, -X, M-1+X, -1, 0, N+1); return res; } __int128_t modified_floor_sum128(__int128_t n, __int128_t m, __int128_t a, __int128_t b, __int128_t x, __int128_t y, __int128_t c){ __int128_t res = 0; __int128_t h, d, w, aa, bb; if(a < 0){ return modified_floor_sum128(n, m, -a, b + (n+1)*a, -x, y, c+x*(n+1)); } h = fDiv(b, m); if(h){ res += (h * n * ( (h+1)*y + (n+1)*x )) / 2 + h * n * c; b -= m * h; c += y * h; } d = fDiv(a, m); res += n * (n+1) / 2 * d * ( 6*c + y*(2*d*n+d+3) + (4*n+2) * x ) / 6; a -= d * m; x += d * y; if(a == 0) return res; w = fDiv(a*n+b, m); aa = -m; bb = b + a * (n+1); res += modified_floor_sum128(w, a, aa, bb, y, -x, c+x*(n+1)); return res; } __int128_t solve128(){ __int128_t i, res = 0; res += modified_floor_sum128(N, M, X, Y-X, 2, 0, -1-N); res += modified_floor_sum128(N, M, -X, M-1+X, -1, 0, N+1); return res; } { Rand rnd; while(0){ ll n, m, a, b, x, y, c; ll r1, r2; n = rnd.get(1,100); m = rnd.get(1,100); a = rnd.get(-100,100); b = rnd.get(-100,100); x = rnd.get(-100,100); y = rnd.get(-100,100); c = rnd.get(-100,100); r1 = modified_floor_sum(n, m, a, b, x, y, c); r2 = baka_modified_floor_sum(n, m, a, b, x, y, c); wt(n,m,a,b,x,y,c); wt(r1,r2); assert(r1==r2); } while(0){ N = rnd.get(1, 100); M = rnd.get(1, 100); X = rnd.get(0, (int)M-1); Y = rnd.get(0, (int)M-1); vector ans; ans.push_back(baka1()); ans.push_back(baka2()); ans.push_back(baka3()); ans.push_back(baka4()); ans.push_back(baka5()); ans.push_back(baka6()); ans.push_back(baka7()); wt(N, M, X, Y); wt(ans); if(Distinct(ans) != 1) return 0; } while(0){ N = rnd.get(1, 1d6); M = rnd.get(1, 1d6); X = rnd.get(0, (int)M-1); Y = rnd.get(0, (int)M-1); __int128_t r1, r2; r1 = solve(); r2 = solve128(); wt(N, M, X, Y); wt(r1,r2); if(r1 != r2) return 0; } T = cReader_ll(1, 1d4, '\n'); rep(T){ ll res; N = cReader_ll(1, 1d9, ' '); M = cReader_ll(1, 1d9, ' '); X = cReader_ll(0, M-1, ' '); Y = cReader_ll(0, M-1, '\n'); res = solve128(); wt(res); } cReader_eof(); }