147 lines
6.6 KiB
Text
147 lines
6.6 KiB
Text
---
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id: intro-nt
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title: "Introductory Number Theory"
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author: Darren Yao, Michael Cao
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prerequisites:
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- Gold - Introduction to Dynamic Programming
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description: Divisibility and Modular Arithmetic
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frequency: 1
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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sample: [
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new Problem("CF", "VK Cup Wildcard R1 C - Prime Factorization", "problemset/problem/162/C", "Intro", false, []),
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],
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general: [
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new Problem("Gold", "Cow Poetry", "897", "Normal", false, ["Knapsack", "Exponentiation"], "First consider the case where there are only two lines with the same class."),
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new Problem("Gold", "Exercise", "1043", "Normal", false, ["Knapsack", "Prime Factorization"], "Prime factorize $K$."),
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],
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}
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};
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## Additional Resources
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<resources>
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<resource source="CPH" title="21.1 - Primes and factors"> </resource>
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<resource source="CPH" title="21.2 - Modular Arithmetic"> </resource>
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<resource source="PAPS" title="16 - Number Theory">same as below</resource>
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<resource source="CF" title="CodeNCode - Number Theory Course" url="blog/entry/77137">lots of advanced stuff you don't need to know at this level</resource>
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</resources>
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## Prime Factorization
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<problems-list problems={metadata.problems.sample} />
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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**.
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Every number has a unique **prime factorization**: a way of decomposing it into a product of primes, as follows:
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$$
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n = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}
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$$
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where the $p_i$ are distinct primes and the $a_i$ are positive integers.
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Now, we will discuss how to find the prime factorization of an integer.
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(pseudocode)
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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.
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Let's look at an example of how this algorithm works, for $n = 252$.
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(table)
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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 $\{2, 2, 3, 3, 7\}$.
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## GCD & LCM
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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 numbers, we use the Euclidean Algorithm, which is as follows:
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$$
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\gcd(a, b) = \begin{cases}
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a & b = 0 \\
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\gcd(b, a \bmod b) & b \neq 0 \\
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\end{cases}
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$$
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This algorithm is very easy to implement using a recursive function in Java, as follows:
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```java
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public int gcd(int a, int b){
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if(b == 0) return a;
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return gcd(b, a % b);
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}
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```
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For C++, use the built-in `__gcd(a,b)`. Finding the GCD of two numbers can be done in $O(\log n)$ time, where $n = \min(a, b)$.
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The **least common multiple (LCM)** of two integers $a$ and $b$ is the smallest integer divisible by both $a$ and $b$.
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The LCM can easily be calculated from the following property with the GCD:
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$$
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\operatorname{lcm}(a, b) = \frac{a \cdot b}{\gcd(a, b)}.
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$$
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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,
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$$
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\gcd(a_1, a_2, a_3, a_4) = \gcd(a_1, \gcd(a_2, \gcd(a_3, a_4))).
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$$
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## Modular Arithmetic
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In **modular arithmetic**, instead of working with integers themselves, we work with their remainders when divided by $m$. We call this taking modulo $m$. For example, if we take $m = 23$, then instead of working with $x = 247$, we use $x \bmod 23 = 17$. Usually, $m$ will be a large prime, given in the problem; the two most common values are $10^9 + 7$, and $998\,244\,353$. Modular arithmetic is used to avoid dealing with numbers that overflow built-in data types, because we can take remainders, according to the following formulas:
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```
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todo no support for gather
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$$
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\begin{gather*}
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(a+b) \bmod m = (a \bmod m + b \bmod m) \bmod m \\
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(a-b) \bmod m = (a \bmod m - b \bmod m) \bmod m \\
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(a \cdot b) \pmod{m} = ((a \bmod m) \cdot (b \bmod m)) \bmod m \\
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a^b \bmod {m} = (a \bmod m)^b \bmod m
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\end{gather*}
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$$
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```
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### Modular Exponentiation
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**Modular Exponentiation** can be used to efficently compute $x ^ n \mod m$. To do this, let's break down $x ^ n$ into binary components. For example, $5 ^ {10}$ = $5 ^ {1010_2}$ = $5 ^ 8 \cdot 5 ^ 4$. Then, if we know $x ^ y$ for all $y$ which are powers of two ($x ^ 1$, $x ^ 2$, $x ^ 4$, $\dots$ , $x ^ {\lfloor{\log_2n} \rfloor}$, we can compute $x ^ n$ in $\mathcal{O}(\log n)$. Finally, since $x ^ y$ for some $y \neq 1$ equals $2 \cdot x ^ {y - 1}$, and $x$ otherwise, we can compute these sums efficently. To deal with $m$, observe that modulo doesn't affect multiplications, so we can directly implement the above "binary exponentiation" algorithm while adding a line to take results $\pmod m$.
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Here is C++ code to compute $x ^ n \pmod m$:
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(not tested)
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```cpp
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long long binpow(long long x, long long n, long long m) {
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x %= m;
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long long res = 1;
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while (n > 0) {
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if (n % 2 == 1) //if n is odd
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res = res * x % m;
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x = x * x % m;
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n >>= 1; //divide by two
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}
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return res;
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}
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```
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### Modular Inverse
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Under a prime modulus, division can be performed using modular inverses.
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To find the modular inverse of some number, simply raise it to the power of $\mathrm{MOD} - 2$, where $\mathrm{MOD}$ is the modulus being used, using the function described above. (The reasons for raising the number to $\mathrm{MOD} - 2$ can be explained with Euler's theorem, but we will not explain the theorem here).
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The modular inverse is the equivalent of the reciprocal in real-number arithmetic; to divide $a$ by $b$, multiply $a$ by the modular inverse of $b$.
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Because it takes $\mathcal{O}(\log \mathrm{MOD})$ time to compute a modular inverse, frequent use of division inside a loop can significantly increase the running time of a program. If the modular inverse of the same number(s) is/are being used many times, it is a good idea to precalculate it.
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Also, one must always ensure that they do not attempt to divide by 0. Be aware that after applying modulo, a nonzero number can become zero, so be very careful when dividing by non-constant values.
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## Problems
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<problems-list problems={metadata.problems.general} />
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