【数论】SSL_1249 C

题意

求出$1\sim N$中有多少个无序数对$(a,b)$的$gcd(a,b)=axorb。$

思路

设$c=gcd(a,b)=axorb$
$\because axorb=c$
$\therefore axorc=b$
$\therefore gcd(a,axorc)=gcd(a,b)=axorb=c$
$\because gcd(a,b)\le a-b,axorb\ge a-b$
$\therefore c=a-b$
$\therefore b=a-c$
$\therefore gcd(a,a-c)=gcd(a,b)=c$
$\because gcd(a,axorc)=c$
$\therefore axorc=a-c$
由设中我们知道$c$为$a$的约数，所以枚举$c$再找出它的倍数$a$。

代码

#include<cstdio>
#include<algorithm>
using namespace std;

int N, ans;

int main() {
	scanf("%d", &N);
	for (int i = 1; i <= N / 2; i++)
		for (int j = i * 2; j <= N; j += i)
			if ((j - i) == (i ^ j)) ans++;
	printf("%d", ans);
}