The Architecture of Integers

🔢 Formal Definition

In the realm of mathematics, an integer ${n,}$ is considered divisible by a non-zero integer ${m,}$ if there exists an integer ${k,}$ such that ${n=km.}$. This relationship is formally denoted as ${m\u2223n.}$

In such cases, ${m,}$ is referred to as a divisor or factor of ${n,}$, and ${n,}$ is a multiple of ${m.}$

⚖️ Conventions on Zero

A point of convention arises regarding whether the divisor ${m,}$ is permitted to be zero.

• Under the convention where ${m,}$ can be zero, ${m\u22230}$ holds for every integer ${m.}$
• Conversely, if ${m}$ must be non-zero, then ${m\u22230}$ holds for every non-zero integer ${m.}$

The definition of divisibility is typically applied to non-zero divisors to avoid trivial cases.

4️⃣2️⃣ Divisors of 42

The integer 7 is a divisor of 42 because ${7 \times 6 = 42,}$ which can be expressed as ${7 \mid 42.}$. The positive divisors of 42 are {1, 2, 3, 6, 7, 14, 21, 42}.

6️⃣ Proper Divisors

The non-trivial divisors of 6 are {2, -2, 3, -3}. A positive divisor of ${n}$ that is not equal to ${n}$ is termed a proper divisor or aliquot part. For 6, the proper divisors are {1, 2, 3}.

6️⃣0️⃣ Divisibility Lattice

The set of positive divisors of an integer, when ordered by divisibility, forms a mathematical structure known as a lattice. For example, the divisors of 60 ({1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}) exhibit this lattice property, where the greatest common divisor (GCD) acts as the meet operation and the least common multiple (LCM) acts as the join operation.

🧮 Number Theory Fundamentals

The study of divisors is intrinsically linked to core concepts in number theory, including integer factorization, prime and composite numbers, and the fundamental theorem of arithmetic, which states that every integer greater than one is either a prime number itself or can be represented as a unique product of prime numbers.

📈 Arithmetic Functions

Functions that operate on integers and relate to their divisors, such as the divisor function ${d(n)}$ (counting divisors) and the sum of divisors function ${\sigma(n)}$, are fundamental tools in number-theoretic investigations.

➗ Properties of Divisibility

Key properties include transitivity (if ${a \mid b}$ and ${b \mid c}$ implies ${a \mid c}$), linearity (if ${a \mid b}$ and ${a \mid c}$, then ${a \mid (b+c)}$), and Euclid's Lemma (if a prime ${p}$ divides the product ${ab}$, then ${p \mid a}$ or ${p \mid b}$). These properties are foundational for understanding number theory.

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