The Timpson Lyceum · Discrete Math (2025)

Number Theory, GCD, and Modular Arithmetic - Lecture 19 (Discrete Math)

Overview

This lecture builds the core machinery of elementary number theory. Starting from the Division Algorithm (every pair of integers has a unique quotient and remainder), we define the greatest common divisor (GCD) and develop Euclid's algorithm, an astonishingly fast way to compute it — roughly 7,000 steps for numbers as large as $10^{100}$, versus the hopeless task of testing every candidate divisor. We then prove that the GCD of $a$ and $b$ is the smallest positive value of $ax + by$, which leads to the Extended Euclidean algorithm for finding those coefficients $x$ and $y$. Finally, we introduce modular arithmetic on the finite sets $\mathbb{Z}_n$ — clock-style addition, multiplication, and subtraction — and see why division is the one operation that resists an easy definition.

Key Concepts

Division Algorithm (recap). For any integers $a$ and $b$ (with $b > 0$), there exist a unique quotient $q$ and remainder $r$ with $a = qb + r, \qquad 0 \le r < b.$ The remainder must satisfy $0 \le r < b$; you can't say "$10$ goes into $27$ once remainder $17$" — the correct statement is twice remainder $7$.

Divisor and common divisor. If $d \mid a$ and $d \mid b$, then $d$ is a common divisor of $a$ and $b$. Example: $1, 2, 4$ are the positive common divisors of $16$ and $28$.

Greatest common divisor. $d = \gcd(a, b)$ if $d$ is a common divisor of $a$ and $b$ and, for every other common divisor $e$, we have $e \le d$. We treat $\gcd$ as a function: $\gcd(16, 28) = 4$.

The key fact behind Euclid's algorithm. Let $c = a \bmod b$ (the remainder). Then $\gcd(a, b) = \gcd(b, c).$ Why it works: writing $a = qb + c$, any divisor of $a$ and $b$ also divides $c = a - qb$, and any divisor of $b$ and $c$ also divides $a = qb + c$. So $\{a,b\}$ and $\{b,c\}$ have exactly the same common divisors, hence the same GCD. (Formally: let $d=\gcd(a,b)$, $e=\gcd(b,c)$; show $d \le e$ and $e \le d$, so $d=e$.)

Euclid's algorithm. Repeatedly replace $(a, b)$ with $(b,\, a \bmod b)$ until the remainder is $0$. The last nonzero remainder is the GCD. This terminates fast because $\gcd(d, 0) = d$ (every integer divides $0$).

GCD as a smallest positive combination. For integers $a, b$ not both zero, the smallest positive value of $ax + by$ over all integers $x, y$ is exactly $\gcd(a, b)$. The set $D = \{ax + by : x,y \in \mathbb{Z},\ ax+by > 0\}$ is a nonempty subset of $\mathbb{N}$ (it contains $a\cdot a + b\cdot b > 0$), so by the well-ordering principle it has a least element $d$; one shows $d \mid a$, $d \mid b$, and that any common divisor $e$ divides the combination $d$, forcing $e \le d$.

Extended Euclidean algorithm. Runs Euclid's algorithm, then back-substitutes to express the GCD as $ax + by$. At each division step solve for the remainder, then work back up, eliminating remainders one at a time until the GCD is written in terms of the original $a$ and $b$.

Relatively prime. Two integers are relatively prime when $\gcd(a, b) = 1$. They need not be prime themselves — e.g. $\gcd(16, 25) = 1$. Consecutive integers are always relatively prime.

Corollary. $a$ and $b$ are relatively prime iff there exist integers $x, y$ with $ax + by = 1$.

Modular arithmetic. Define $\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}$ (for $n > 1$). Arithmetic "wraps around" like a clock that starts at $0$: $a \oplus b = (a + b) \bmod n, \qquad a \otimes b = (a \cdot b) \bmod n.$ Both operations are closed on $\mathbb{Z}_n$ (the remainder is always in range), commutative, associative, with additive identity $0$ and multiplicative identity $1$, and they distribute. ($\mathbb{Z}_n$ under $\oplus$ is a group.)

Congruence. $a \equiv b \pmod{n}$ means $n \mid (a - b)$, which is equivalent to $(a \bmod n) = (b \bmod n)$.

Modular subtraction. $a \ominus b$ is the unique $x \in \mathbb{Z}_n$ with $b \oplus x = a$; it is computed as $(a - b) \bmod n$. For example in $\mathbb{Z}_{12}$, $1 - 3 = -2 \equiv 10$ (3 hours before 1 o'clock is 10 o'clock).

Why division is hard (preview). In $\mathbb{Z}_{10}$, $4 \div 2$ would need an $x$ with $2x = 4$; both $x = 2$ and $x = 7$ work ($2\cdot 7 = 14 \equiv 4$), so the answer isn't unique. A multiplicative inverse exists only when the numbers are relatively prime to $n$ — handled next time.

Worked Example

Extended Euclidean algorithm: find $x, y$ with $689x + 234y = \gcd(689, 234)$.

Run Euclid's algorithm (left), recording each remainder (right):

Back-substitute, eliminating the remainder $221$:

So $689(-1) + 234(3) = 13$, giving $x = -1$, $y = 3$.

Summary

• The Division Algorithm guarantees a unique quotient and remainder with $0 \le r < b$; this underpins everything that follows.
• $\gcd(a, b) = \gcd(b,\, a \bmod b)$, so Euclid's algorithm finds the GCD as the last nonzero remainder — exponentially faster than trial division.
• The GCD of $a$ and $b$ equals the smallest positive value of $ax + by$; the Extended Euclidean algorithm recovers the coefficients $x, y$ by back-substitution.
• Two integers are relatively prime when their GCD is $1$, equivalently when $ax + by = 1$ has an integer solution.
• $\mathbb{Z}_n = \{0, \ldots, n-1\}$ with $a \oplus b = (a+b)\bmod n$ and $a \otimes b = (ab)\bmod n$ is closed, commutative, associative, and has identities $0$ and $1$.
• Modular subtraction is well-defined as $(a-b)\bmod n$; modular division fails in general and works only when the divisor is relatively prime to $n$.