LightOJ 1289 - LCM from 1 to n

You are given T test cases, for each test case you are given n.

You have to find lcm(1, 2,...,n) where lcm refers to the least common multiple of some integers, i.e. the smallest integer that is a multiple of all of the given integers (modulo 2^32).

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Summary

lcm(b₁, b₂, ... , b_m) can be represented as p₁^a₁ . p₂^a₂ . ... . p_k^a_k where p₁, p₂, ... , p_k are prime factors of those numbers and a₁, a₂, ... , a_k are the maximum powers of those corresponding primes, that occur in the numbers b₁, b₂, ... , b_k . This number theory problem requires you to find the prime factors and their corresponding powers and the product of these.

In this problem, you can assume b₁, b₂, ... , b_m = 1,2, ... , n .

Solution

Naive Approach

Let's calculate for each prime (until n) the maximum power a of this prime p, so that p^a <= n. For example, if n = 100, prime 5 has highest power of log₅100, i.e. the maximum number of time we can divide 100 by 5, which is 2. You can see, 5² = 25 which is less than 100, and 5³ = 125 which is greater than 100. So here, for n = 100, p = 5 we get a = 2.

So, for each prime p_i until n, we calculate a_i , which is the maximum power of that prme till n. Then, lcm = p₁^a₁ . p₂^a₂ . ... . p_k^a_k , which is the required answer.

Problem with Naive Solution

n can be equal to 10⁸, and T can be equal to 10⁴ . There are roughly k = 6x10⁶ primes till 10⁸ . So for each case, if we intend to use the naive solution, the complexity for each case would be O(k log n), where k is the number of prime factors till n. So, total complexity will be O(Tk log n) , which won't pass the time limit of 4s.

Observations

• Every prime not greater than n appears in the LCM at least once, i.e. a_i is at least 1.

Why: Since p_i <= n, when we calculate lcm(1, 2, 3, ... , n), p_i is included in 1, 2, 3, ..., n . So it appears atleast once in the LCM.

• No prime greater than n will appear in the LCM.

Why: Let p_i > n is a prime. There is no multiple of p_i that is included in 1, 2, 3, ... , n since the lowest multiple of p_i is p_i itself, which is greater than n.

• Every prime till n and not greater than sqrt(n) will appear atleast twice in the LCM.

Why: Let p_i <= sqrt(n) be a prime not greater than n. Let p_i^a_i appear in the prime. Since p_i <= sqrt(n), p_i² <= n, so a_i is at least 2.

• Every prime till n and greater than sqrt(n) will apear exactly once in the LCM.

Why: Let p_i > sqrt(n) be a prime not greater than n. Let p_i^a_i appear in the prime. Now, p_i > sqrt(n) implies that p_i² > n, so a_i can not be greater than 1. And, according to the first observation, a_i is at least 1. So, a_i is exactly 1, which means p_i appears exactly once in the LCM.

These observation paves the way for some workarounds and optimizations that gives a faster approach.

Faster Approach

• If we know the product of all primes till n, we can consider this product as "taking each prime till n atleast once" for the LCM. So we can precalculate this before processing any of the test cases, for all possible n using techniques like cumulative sum (here we can call it cumulative product).

• For each case, we can find the product of primes till n using a binary search. Taking this product implies we took all the primes till n at least once. So, we don't need to consider primes greater than sqrt(n) according to the fourth observation.

• Now we can only consider primes till sqrt(n). If any prime p_i appears a_i times in n, we multiply p_i , a_i - 1 times with the answer (since we already took these primes once in the previous step).

• Done!

The preprocessing needs O(n) operations to complete, for just once, since we are just running a loop till the max value of n.

For each case, we are considering all primes till sqrt(n) and checking how many times they appear till n. This requires O(sqrt(n) * log2(n)) operations at most.

So, total complexity for T cases = O(n + T * sqrt(n) * log2(n)) , which passes the 4s time limit.

C++ code

#include "bits/stdc++.h"

using namespace std;
using ll = long long;
using uint = unsigned int;

//storing the values in unsigned int
//since it prevents exceeding memory limit
//also, it automatically makes the numbers
//modulo 2^32

const int mx = 1e8 + 9; //max value of n
const int mxprm = 6e6 + 9; //max number of primes

int psz = 0; //keeping count the number of primes discovered

bitset <mx> mark; //to keep track of primes
uint primes[mxprm]; //to store the primes
uint mul[mxprm];  //to store the cumulative product of the primes

void sieve() { //just a prime sieve code
    mark[0] = mark[1] = 1;
    primes[psz++] = 2;
    int lim = sqrt(mx * 1.0) + 2;
    for (int i = 4; i < mx; i += 2) mark[i] = 1;
    for (int i = 3; i < mx; i += 2) {
        if (!mark[i]) {
            primes[psz++] = i;
            if (i <= lim)
                for (int j = i * i; j < mx; j += i)
                    mark[j] = 1;
        }
    }
}

int main() {
    sieve();

    mul[0] = 2; //first prime is 2, so cumulative product till first prime is 2
    for (int i = 1; i < psz; i++)
        mul[i] = (primes[i] * mul[i - 1]); //calculating cumulative product

    int tc; scanf("%d", &tc);
    int kase = 0;
    while (tc--) {
        int n; scanf("%d", &n);
        printf("Case %d: ", ++kase);

        uint ans = 1;

        int idx =  upper_bound(primes, primes + psz, n) - primes;
        //upper_bound gives the iterator to the smallest prime greater than n
        //idx contains it's index

        idx--; //now, idx contains the index of the largest prime not greater than n

        ans *= mul[idx];
        for (int i = 0; i < psz; i++) {
            ll p = primes[i];
            ll x = n;
            ll a = 0;

            if (p * p > n) break; //checking primes till sqrt(n) is enough

            //for each prime p, let's find the maximum power of p till n
            //stored in the variable a
            while (x >= p) {
                x /= p;
                a++;
            }

            ans *= pow(p, a - 1); //multiplying by p^(a-1) since we took these primes once before
        }

        //answer is already calculated modulo 2^32 because of taking unsigned integer
        printf("%lld\n", ans);
    }
}

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Tutorial source