Divisibility
Authors: Darren Yao, Michael Cao, Andi Qu, and Kevin Sheng
Using the information that one integer evenly divides another.
If you've never encountered any number theory before, AoPS is a good place to start.
Resources | ||||
---|---|---|---|---|
AoPS | practice problems, set focus to number theory! | |||
AoPS | good book |
Resources
Resources | ||||
---|---|---|---|---|
IUSACO | module is based off this | |||
David Altizio | ||||
CPH | ||||
PAPS | ||||
MONT | ||||
Sato | nice proofs and problems |
Prime Factorization
Focus Problem – read through this problem before continuing!
A positive integer is called a divisor or a factor of a non-negative integer if is divisible by , which means that there exists some integer such that . An integer is prime if its only divisors are and . Integers greater than that are not prime are composite.
Every positive integer has a unique prime factorization: a way of decomposing it into a product of primes, as follows:
where the are distinct primes and the are positive integers.
Now, we will discuss how to find the prime factorization of any positive integer.
C++
vector<int> factor(int n) {vector<int> ret;for (int i = 2; i * i <= n; i++) {while (n % i == 0) {ret.push_back(i);n /= i;}}if (n > 1) ret.push_back(n);return ret;}
Java
ArrayList<Integer> factor(int n) {ArrayList<Integer> factors = new ArrayList<>();for (int i = 2; i * i <= n; i++) {while (n % i == 0) {factors.add(i);n /= i;}}if (n > 1) factors.add(n);return factors;}
Python
def factor(n):ret = []i = 2while i * i <= n:while n % i == 0:ret.append(i)n //= ii += 1if n > 1:ret.append(n)return ret
This algorithm runs in time, because the for loop checks divisibility for at most values. Even though there is a while loop inside the for loop, dividing by quickly reduces the value of , which means that the outer for loop runs less iterations, which actually speeds up the code.
Let's look at an example of how this algorithm works, for .
At this point, the for loop terminates, because is already 3 which is greater than . In the last step, we add to the list of factors , because it otherwise won't be added, for a final prime factorization of .
Solution - Counting Divisors
The most straightforward solution is just to do what the problem asks us to do - for each , find the number of divisors of in time.
C++
#include <iostream>using namespace std;int main() {int n;cin >> n;for (int q = 0; q < n; q++) {int x;int div_num = 0;
Java
import java.io.BufferedReader;import java.io.IOException;import java.io.InputStreamReader;public class Divisors {public static void main(String[] args) throws IOException {BufferedReader read = new BufferedReader(new InputStreamReader(System.in));int queryNum = Integer.parseInt(read.readLine());StringBuilder ans = new StringBuilder();for (int q = 0; q < queryNum; q++) {
This solution runs in time, which is just fast enough to get AC. However, we can actually speed this up to get an solution!
First, let's discuss an important property of the prime factorization. Consider:
Then the number of divisors of is simply .
Why is this true? The exponent of in any divisor of must be in the range and each different exponent results in a different set of divisors, so each contributes to the product.
can have distinct prime factors, so if we can find the prime factorization of efficiently, we can answer queries in time instead of the previous time.
Here's how we find the prime factorization of in time with preprocessing:
- For each , find any prime number that divides .
- We can use the Sieve of Eratosthenes to find this efficiently.
- For each , we can then find the prime factorization by repeatedly dividing by a prime number that divides until .
Alternatively, we can slightly modify the the prime factorization code above.
C++
#include <iostream>using namespace std;const int MAX_N = 1e6;// max_div[i] contains the largest prime that goes into iint max_div[MAX_N + 1];int main() {for (int i = 2; i <= MAX_N; i++) {
Java
import java.io.BufferedReader;import java.io.IOException;import java.io.InputStreamReader;public class Divisors {private static final int MAX_N = (int) Math.pow(10, 6);public static void main(String[] args) throws IOException {// maxDiv[i] contains the largest prime that can divide iint[] maxDiv = new int[MAX_N + 1];for (int i = 2; i <= MAX_N; i++) {
Python
MAX_N = 10 ** 6# max_div[i] contains the largest prime that can go into imax_div = [0 for _ in range(MAX_N + 1)]for i in range(2, MAX_N + 1):if max_div[i] == 0:for j in range(i, MAX_N + 1, i):max_div[j] = ians = []
Optional
Apply the linear sieve.
GCD & LCM
GCD
The greatest common divisor (GCD) of two integers and is the largest integer that is a factor of both and . In order to find the GCD of two non-negative integers, we use the Euclidean Algorithm, which is as follows:
This algorithm is very easy to implement using a recursive function, as follows:
Java
public int gcd(int a, int b) {return b == 0 ? a : gcd(b, a % b);}
C++
int gcd(int a, int b) {return b == 0 ? a : gcd(b, a % b);}
For C++14, you can use the built-in __gcd(a,b)
.
In C++17, there exists std::gcd
and std::lcm
in the
<numeric>
header, so there's no
need to code your own GCD and LCM if you're using that.
Python
def gcd(a, b):return a if b == 0 else gcd(b, a % b)
This function runs in time because .
The worst-case scenario for the Euclidean algorithm is when and are consecutive Fibonacci numbers and . for an explanation). In this case, the algorithm will calculate . This means that finding takes steps, which is proportional to .
LCM
The least common multiple (LCM) of two integers and is the smallest integer divisible by both and . The LCM can easily be calculated from the following property with the GCD:
Warning!
Coding as a * b / gcd(a, b)
might cause integer overflow if the
value of a * b
is greater than the max size of the data type of a * b
(e.g.
the max size of int
is around 2 billion). Dividng a
by gcd(a, b)
first,
then multiplying it by b
will prevent integer overflow if the result fits in
an int
.
If we want to take the GCD or LCM of more than two elements, we can do so two at a time, in any order. For example,
Problems
Status | Source | Problem Name | Difficulty | Tags | |
---|---|---|---|---|---|
AC | Easy | Show TagsPrime Factorization | |||
CF | Easy | Show TagsDivisibility, Modular Arithmetic | |||
CF | Easy | Show TagsNT | |||
CF | Easy | Show TagsDivisibility | |||
CSES | Normal | ||||
CF | Normal | Show TagsPrime Factorization | |||
CSES | Hard | ||||
CF | Hard | Show TagsDivisibility |