Principle of Co-Prime exclusion in HDU-4135

Given a number, this question is used to determine how many digits in a certain range interact with this number.

First, we need to make it clear that if we can find the number of all numbers that are not in mutual quality in a given interval, then the number of mutual quality numbers will be obtained. This is called reverse solution.

For any positive integer N, there is a unique prime factor decomposition form. We can think that any number that does not interact with N must be a multiple of a certain prime factor, therefore, we can use a quality factor to determine the number of numbers in a range and the number of multiples of the quality factor. However, this is not a sufficient condition for all non-prime numbers. For other qualitative factors that can determine a series of numbers, what we need to do is to combine the over-capacity exclusion theorem to obtain their union.

CodeAs follows:

# Include <cstdlib> # Include <Cstring> # Include <Cstdio> # Include <Cmath> Using   Namespace  STD; typedef  Long   Long  Int  Int64; int64 a, B, num, REC [  6000000  ], Lim, ans;  Int  Idx; int64 gcd (int64 A, int64 B) {int64 T;  While  (B) {T = A; = B; B = T % B ;}  Return  A;} int64 lcm (int64 A, int64 B ){  Return A/gcd (a, B )* B ;}  Void DFS ( Int P, int64 Co, Int  Sign ){  If (P> Idx ){  Return  ;}  If (P> 0  ) {Ans + = Sign * (B/Co-(- 1 )/ CO );} For ( Int I = P + 1 ; I <= idx; ++ I) {DFS (I, lcm (CO, REC [I]), - Sign );}}  Int  Main (){  Int T, CA = 0  ; Scanf (  "  % D  " ,& T ); While (T -- ) {Idx = 0  ; Ans = 0  ; Scanf (  "  % I64d % i64d % i64d  " , & A, & B ,& Num );  If (Num = 1  ) {Printf (  " Case # % d: % i64d \ n  " , ++ Ca, B-A + 1  );  Continue  ;} Lim = ( Int ) SQRT ( Double  (Num ));  If (Num % 2 = 0  ){  While (Num %2 = 0  ) {Num >>= 1  ;} Rec [ ++ Idx] = 2  ;}  For ( Int I = 3 ; I <= lim; ++ I ){  If (Num % I = 0  ){ While (Num % I = 0  ) {Num /= I;} rec [ ++ Idx] = I ;}}  If (Num! = 1  ) {Rec [ ++ Idx] = Num;} DFS (  0 , 1 ,- 1 ); Printf (  "  Case # % d: % i64d \ n  " , ++ Ca, B-A + 1 - Ans );}  Return   0  ;}