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  • LightOJ 1289 LCM from 1 to n

    1289 - LCM from 1 to n

    Given an integer n, you have to find

    lcm(1, 2, 3, ..., n)

    lcm means least common multiple. For example lcm(2, 5, 4) = 20, lcm(3, 9) = 9, lcm(6, 8, 12) = 24.

    Input

    Input starts with an integer T (≤ 10000), denoting the number of test cases.

    Each case starts with a line containing an integer n (2 ≤ n ≤ 108).

    Output

    For each case, print the case number and lcm(1, 2, 3, ..., n). As the result can be very big, print the result modulo 232.

    Sample Input

    Output for Sample Input

    5

    10

    5

    200

    15

    20

    Case 1: 2520

    Case 2: 60

    Case 3: 2300527488

    Case 4: 360360

    Case 5: 232792560

    Solution:

    首先,lcm(1,2,...,n)为2k1*3k2*...*pxkx,pi为小于等于n的质数,piki<=n且pi(ki+1)>n

    所以,先预处理出1e8范围内的质数,及其前缀积.

    设pro[i]为∏p[j](1<=j<=i)

    ans=1;
    for(i=1;i<=31;++i)
    {
      找到p[j]i<=n且p[j+1]i>n;
      ans*=pro[j];
    }

    Code:

    #include<iostream>
    #include<algorithm>
    #include<cstdio>
    #include<cmath>
    #include<cstring>
    #include<bitset>
    using namespace std;
    #define uint unsigned int
    #define ull unsigned long long
    const uint MAXN=1e8;
    const uint MAXTOT=5761455;
    const uint MAXPOW=31;
    const uint INF=0xffffffff;
    uint t,CASE;
    bitset<MAXN+10> vis;
    uint p[MAXTOT+10],tot;
    uint pro[MAXTOT+10];
    void GetP()
    {
    	for(uint i=2;i<=MAXN;++i)
    	{
    		if(!vis[i])p[++tot]=i;
    		for(uint j=1;j<=tot&&(ull)p[j]*i<=MAXN;++j)
    		{
    			vis[p[j]*i]=true;
    			if(i%p[j]==0)break;
    		}
    	}
    }
    uint QuickPow(uint x,uint y)
    {
    	ull res=1;
    	while(y--)
    	{
    		res*=x;
    		if(res>INF)return INF;
    	}
    	return res;
    }
    void init()
    {
    	GetP();
    	pro[0]=1;
    	for(uint i=1;i<=tot;++i)pro[i]=pro[i-1]*p[i];
    }
    uint n;
    int main()
    {
    	init();
    	scanf("%u",&t);
    	while(t--)
    	{
    		scanf("%u",&n);
    		uint res=1;
    		for(uint i=1;i<=MAXPOW;++i)
    		{
    			uint left=0,mid,right=MAXTOT,r=0;
    			while(left<=right)
    			{
    				mid=(left+right)>>1;
    				if(QuickPow(p[mid],i)<=n)
    				{
    					r=mid;
    					left=mid+1;
    				}
    				else right=mid-1;
    			}
    			res*=pro[r];
    		}
    		printf("Case %d: %u
    ",++CASE,res);
    	}
    	return 0;
    }
    
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  • 原文地址:https://www.cnblogs.com/DOlaBMOon/p/7324846.html
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