Codeforces 710 D. Arithmetic progressions

Description

\ (x=a_1k+b_1=a_2l+b_2,l\leqslant x \leqslant r\) the number of \ (x\) that satisfies such conditions.

Sol

Expand Euclid + Chinese remainder theorem.

Find this equivalent to a linear equation set.

\ (x \equiv b_1 (mod a_1) \)

\ (x \equiv b_2 (mod a_2) \)

Subtract from the original two (a_1k-a_2l=b_2-b_1\)

This is used to expand Euclid to seek, if \ ((a_1,a_2) \nmid (b_2-b_1) \) obviously no solution.

The equation used to expand Euclid is \ (a_1k-a_2l= (a_1,a_2) \), and multiply the equation by \ (\frac{b_2-b_1}{(a_1,a_2)}\)

Now we have a set of legal solutions, the general solution is \ (K=k_0+\frac {a_2}{(a_1,a_2)},l=l_0-\frac {a_1}{(a_1,a_2)}\)

The minimum positive solution can be modeled on \ (\frac {a_2}{(a_1,a_2)}\).

Then the solution number of the original equation is \ (k+n[a_1,a_2]\), do not forget to calculate this value of the endpoint.

Code

#include <cstdio> #include <algorithm> #include <iostream>using namespace std;typedef long Long ll;# Define Debug (a) cout<< #a << "=" <<a<< "ll a1,b1,a2,b2,k,l,x,lcm,gcd,l,r,ans;void exgcd (ll A,ll B,ll &x,ll &y) {if (!b) {X=1,y=0;return;} EXGCD (B,a%b,x,y); LL t=x;x=y,y=t-(A/b) *y;} int main () {Ios::sync_with_stdio (false);cin>>a1>>b1>>a2>>b2>>l>>r; EXGCD (a1,a2,k,l); GCD=__GCD (A1,A2), LCM=A1/GCD*A2; L=max (L,max (B1,B2)); if (b2-b1)%GCD | | L>r) return puts ("0"), 0;k*= (B2-B1)/gcd,k= (k% (A2/GCD) +a2/gcd)% (A2/GCD), X=a1*k+b1;//debug (x), Debug (LCM), Debug (L ), Debug (R), if (r>=x) ans+= (r-x)/lcm+1;if (l-1>=x) ans-= (l-1-x)/lcm+1;cout<<ans<<endl;return 0;}

　　

Codeforces 710 D. Arithmetic progressions