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# Divisibility

Authors: Darren Yao, Michael Cao, Andi Qu

Using the information that A is a factor of B

If you've never encountered any number theory before, AoPS is a good place to start.

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## Resources

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IUSACOmodule is based off this
David Altizio
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## Prime Factorization

Focus Problem – read through this problem before continuing!

A number $a$ is called a divisor or a factor of a number $b$ if $b$ is divisible by $a$, which means that there exists some integer $k$ such that $b = ka$. Conventionally, $1$ and $n$ are considered divisors of $n$. A number $n > 1$ is prime if its only divisors are $1$ and $n$. Numbers greater than (1) that are not prime are composite.

Every number has a unique prime factorization: a way of decomposing it into a product of primes, as follows:

$n = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}$

where the $p_i$ are distinct primes and the $a_i$ are positive integers.

Now, we will discuss how to find the prime factorization of an integer.

C++

1vector<int> factor(int n) {2    vector<int> ret;3    for (int i = 2; i * i <= n; i++) {4        while (n % i == 0) {5            ret.push_back(i);6            n /= i;7        }8    }9    if (n > 1) ret.push_back(n);10    return ret;

Java

1ArrayList<Integer> factor(int n) {2    ArrayList<Integer> factors = new ArrayList<>();3    for (int i = 2; i * i <= n; i++) {4        while (n % i == 0) {5            factors.add(i);6            n /= i;7        }8    }9    if (n > 1) factors.add(n);10    return factors;

Python

1def factor(n):2    ret = []3    i = 24    while i * i <= n:5        while n % i == 0:6            ret.append(i)7            n //= i8        i += 19    if n > 1:10        ret.append(n)

This algorithm runs in $O(\sqrt{n})$ time, because the for loop checks divisibility for at most $\sqrt{n}$ values. Even though there is a while loop inside the for loop, dividing $n$ by $i$ quickly reduces the value of $n$, 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 $n = 252$.

At this point, the for loop terminates, because $i$ is already 3 which is greater than $\lfloor \sqrt{7} \rfloor$. In the last step, we add $7$ to the list of factors $v$, because it otherwise won't be added, for a final prime factorization of $\{2, 2, 3, 3, 7\}$.

### Solution - Counting Divisors

The most straightforward solution is just to do what the problem asks us to do - for each $x$, find the number of divisors of $x$ in $O(\sqrt x)$ time.

C++

1#include <bits/stdc++.h>2using namespace std;3 4int main() {5    ios_base::sync_with_stdio(0);6    cin.tie(0);7    int n;8    cin >> n;9    while (n--) {10        int x, ans = 0;

Java

1import java.io.*;2import java.util.*;3
4public class Main {5    static int solve(int n) {6        int divisors = 0;7        for (int i = 1; i * i <= n; i++) {8            if (n % i == 0) {9                if (i * i == n) divisors++;10                else divisors += 2;

This solution runs in $O(n \sqrt x)$ time, which is just fast enough to get AC. However, we can actually speed this up to get an $O((x + n) \log x)$ solution!

First, let's discuss an important property of the prime factorization of a number. Consider:

$x = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}$

Then the number of divisors of $x$ is simply $(a_1 + 1) \cdot (a_2 + 1) \cdots (a_k + 1)$.

Why is this true? The exponent of $p_i$ in any divisor of $x$ must be in the range $[0, a_i]$ and each different exponent results in a different set of divisors, so each $p_i$ contributes $a_i + 1$ to the product.

$x$ can have $O(\log x)$ distinct prime factors, so if we can find the prime factorization of $x$ efficiently, we can answer queries in $O(\log x)$ time instead of the previous $O(\sqrt x)$ time.

Here's how we find the prime factorization of $x$ in $O(\log x)$ time with $O(x \log x)$ preprocessing:

• For each $k \leq 10^6$, find any prime number that divides $k$.
• We can use the Sieve of Eratosthenes to find this efficiently.
• For each $x$, we can then find the prime factorization by repeatedly dividing $x$ by a prime number that divides $x$ until $x = 1$.

Alternatively, we can slightly modify the the prime factorization code above.

C++

1#include <bits/stdc++.h>2using namespace std;3
4int max_div;5
6int main() {7    ios_base::sync_with_stdio(0);8    cin.tie(0);9    for (int i = 2; i <= 1000000; i++) {10        if (!max_div[i]) {

Java

### This section is not complete.

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make java code consistent w/ C++ code ...

1import java.io.*;2import java.util.*;3
4public class Main {5    static int solve(int n) {6        int divisors = 1;7        for (int i = 2; i * i <= n; i++) {8            int ct = 0;9            while (n % i == 0) {10                ct++;
Optional

Apply the linear sieve.

## GCD & LCM

### GCD

The greatest common divisor (GCD) of two integers $a$ and $b$ is the largest integer that is a factor of both $a$ and $b$. In order to find the GCD of two non-negative integers, we use the Euclidean Algorithm, which is as follows:

$\gcd(a, b) = \begin{cases} a & b = 0 \\ \gcd(b, a \bmod b) & b \neq 0 \\ \end{cases}$

This algorithm is very easy to implement using a recursive function, as follows:

Java

1public int gcd(int a, int b){2    if (b == 0) return a;3    return gcd(b, a % b);4}

C++

1int gcd(int a, int b){2    if (b == 0) return a;3    return gcd(b, a % b);4}

For C++14, you can use the built-in __gcd(a,b).

Python

1def gcd(a, b):2    if b == 0:3        return a4    return gcd(b, a % b)

This function runs in $O(\log ab)$ time because $a\le b \implies b\%a <\frac{b}{2}$.

The worst-case scenario for the Euclidean algorithm is when $a$ and $b$ are consecutive Fibonacci numbers $F_n$ and $F_{n + 1}$. for an explanation). In this case, the algorithm will calculate $\gcd(F_n, F_{n + 1}) = \gcd(F_{n - 1}, F_n) = \dots = \gcd(0, F_1)$. This means that finding $\gcd(F_n, F_{n + 1})$ takes $n + 1$ steps, which is proportional to $\log \left(F_n F_{n+1}\right)$.

### LCM

The least common multiple (LCM) of two integers $a$ and $b$ is the smallest integer divisible by both $a$ and $b$. The LCM can easily be calculated from the following property with the GCD:

$\operatorname{lcm}(a, b) = \frac{a \cdot b}{\gcd(a, b)}=\frac{a}{\gcd(a,b)}\cdot b.$

### Warning!

Coding $\text{lcm}$ 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,

$\gcd(a_1, a_2, a_3, a_4) = \gcd(a_1, \gcd(a_2, \gcd(a_3, a_4))).$

Exercise: What's wrong with the following code?

1ll gcd(ll a, ll b){ return b == 0 ? a : gcd(b,a%b); }2ll lcm(ll a, ll b) { return a/gcd(a,b)*b; }3
4int main() { cout << lcm(1000000000,999999999); } // output: 1808348672

Solution

## Problems

StatusSourceProblem NameDifficultyTagsSolution
ACEasy
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Prime Factorization

Check AC
CSESNormal
CFNormal
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Prime Factorization

CSESHard