HDU1695 GCD

タイトルの再現
Given 5 integers: a,b,c,d,k , you’re to find x∈[a,b],y∈[c,d] that gcd(x,y)=k . gcd(x,y) means the greatest common divisor of x and y . Since the number of choices may be very large, you’re only required to output the total number of different number pairs. Please notice that, (x=5,y=7) and (x=7,y=5) are considered to be the same.
You can assume that a=c=1 in all test cases.
Input
The input consists of several test cases. The first line of the input is the number of the cases. There are no more than 3,000 cases. Each case contains five integers: a,b,c,d,k,0Output
For each test case, print the number of choices. Use the format in the example.
Sample Input

2
1 3 1 5 1
1 11014 1 14409 9

Sample Output

Case 1: 9
Case 2: 736427

問題解
問題は簡単な結論に基づいています.
∑1≤x≤A,1≤y≤B[K|gcd(x,y)]=⌊AK⌋⌊BK⌋
セット
GAB(K)=∑1≤x≤A,1≤y≤B[K|gcd(x,y)]FAB(K)=∑1≤x≤A,1≤y≤B[K=gcd(x,y)]
次のように表示されます.
GAB(K)=∑K|kFAB(k)
モビウスの反転原理には、
FAB(k)=∑d≥1μ(d)GAB(dk)
したがって、
FAB(K)=∑1≤x≤A,1≤y≤B[K=gcd(x,y)]=∑d≥1μ(d)⌊AdK⌋⌊BdK⌋
この問題では要求(x,y),(y,x)はカウントを繰り返さない
A≦Bとしてもよい
Ans(A,B,K)=FAB(K)−FAA(K)2
リファレンスコード

HDU 1695モビウス反転