cf 833 A 数论
Slastyona and her loyal dog Pushok are playing a meaningless game that is indeed very interesting.
The game consists of multiple rounds. Its rules are very simple: in each round, a natural number k is chosen. Then, the one who says (or barks) it faster than the other wins the round. After that, the winner's score is multiplied by k2, and the loser's score is multiplied by k. In the beginning of the game, both Slastyona and Pushok have scores equal to one.
Unfortunately, Slastyona had lost her notepad where the history of all n games was recorded. She managed to recall the final results for each games, though, but all of her memories of them are vague. Help Slastyona verify their correctness, or, to put it another way, for each given pair of scores determine whether it was possible for a game to finish with such result or not.
In the first string, the number of games n (1 ≤ n ≤ 350000) is given.
Each game is represented by a pair of scores a, b (1 ≤ a, b ≤ 109) – the results of Slastyona and Pushok, correspondingly.
For each pair of scores, answer "Yes" if it's possible for a game to finish with given score, and "No" otherwise.
You can output each letter in arbitrary case (upper or lower).
6
2 4
75 45
8 8
16 16
247 994
1000000000 1000000
Yes
Yes
Yes
No
No
Yes
First game might have been consisted of one round, in which the number 2 would have been chosen and Pushok would have won.
The second game needs exactly two rounds to finish with such result: in the first one, Slastyona would have said the number 5, and in the second one, Pushok would have barked the number 3.
题意，给出两个人的初始分值都是1，和结束分值(a,b),现在判断有没有可能通过数局游戏到达这个分值。
规则，每次选出一个自然数k，其中一个人的分值乘上k*k,另一个人就乘上k，反之亦然。
当时推出来式子了，却没想到怎么证明哎。
设进行了n局游戏，则有 a*b=(k1*k2*k3......kn)3,这个并不难证明，我们假设存在整数c=k1*k2*k3....*kn使得等式成立，
则c=cbrt(a*b),接着就要找c和a,b的关系，如果c真的存在那么a,b都能整除以c，x=a/c,y=b/c;
如果x,y是正确的解那么代回去之后 a=x*x*y b=y*y*x; 判断一下就好了。
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define LL long long 4 int main() 5 { 6 int n; 7 LL a,b; 8 scanf("%d",&n); 9 while(n--){ 10 scanf("%lld%lld",&a,&b); 11 LL c=cbrt((long double)a*b); 12 LL x=a/c,y=b/c; 13 if(a==x*x*y&&b==y*y*x) puts("Yes"); 14 else puts("No"); 15 } 16 return 0; 17 }
上面是看的别人的其实这个思路不是很好懂，如果c存在的话，那么我们可以二分出c的值进行判定c*c*c==a*b是否成立即可，但注意这并不是充要条件，
c还要满足 a%c==0&&b%c==0没写这两个导致我WA
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define LL long long 4 LL solve(LL a,LL b) 5 { 6 LL l=0,r=1e6; 7 while(l<r){ 8 LL mid=(l+r)>>1; 9 LL m3=mid*mid*mid; 10 if(m3==a*b&&a%mid==0&&b%mid==0) return 1; 11 else if (m3>a*b) r=mid-1; 12 else l=mid+1; 13 } 14 if(l==r&&l*l*l==a*b&&a%l==0&&b%l==0) return 1; 15 return 0; 16 } 17 int main() 18 { 19 int n; 20 LL a,b; 21 scanf("%d",&n); 22 while(n--){ 23 scanf("%lld%lld",&a,&b); 24 if(solve(a,b)) puts("Yes"); 25 else puts("No"); 26 } 27 return 0; 28 }
