PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Number Theory → Elementary Number Theory ↓


Divisibility


Divisibility is one of the fundamental concepts in elementary number theory, a branch of mathematics that deals with the properties and relationships of numbers. It forms the basis for a wide range of mathematical ideas and is essential for understanding more advanced topics. In this lesson, we will delve deeper into the concept of divisibility, exploring its definitions, properties, theorems, and applications with plenty of examples and explanations.

Understanding divisibility

In simple terms, a number a is said to be divisible by another number b if a can be divided by b without leaving a remainder. We often write this relation as:

a = b × k

where k is an integer. If such an integer k exists, then we say that b divides a, and we write it as:

B | A

For example, consider the numbers 10 and 5. We can say that 5 divides 10 because:

10 = 5 × 2

In this example, k = 2, which is an integer, and thus the statement 5 | 10 is true.

Divisibility properties

There are many important properties of divisibility that can be very useful in problem-solving. Here, we list and explain some of these properties:

  • Reflexive property: Any integer a is divisible by itself. Mathematically, we express this as:
    one | one
    This is because a = a × 1.
  • Transitive property: If a | b and b | c, then a | c. In other words, divisibility is transitive. For example, if 2 divides 4, and 4 divides 12, then 2 divides 12.
  • Multiplication property: If a | b, then for any integer c, a | (b × c). This means that the number b multiplied by any integer c also gives a multiple of a.
  • Additive property: If a | b and a | c, then a | (b + c) This shows that if a divides two numbers, it also divides their sum.

Divisibility hypothesis

Let's try to understand the concept of divisibility using a simple example. Consider the number 12 and its divisors.

12 = 1 × 12
12 = 2 × 6
12 = 3 × 4

These divisions are represented here:


    
    1
    
    2
    
    3
    
    4
    
    
    6
    
    12

These boxes represent the factors of 12, and clearly represent the pairs of numbers that divide 12 without a remainder: 1 × 12, 2 × 6, and 3 × 4.

Normal divisibility test

There are some quick tests or rules that can help determine whether a number is divisible by another number. These tests often involve arithmetic tricks.

Divisibility by 2

A number is divisible by 2 if it ends in an even digit (0, 2, 4, 6, 8). For example, 14, 26, and 38 are all divisible by 2.

Divisibility by 3

To check if a number is divisible by 3, sum all its digits. If the result is divisible by 3, then the number is also divisible by 3. For example, for the number 123, we have:

1 + 2 + 3 = 6

Since 6 is divisible by 3, 123 is also divisible.

Divisibility by 5

If a number ends in 0 or 5 then it is divisible by 5. For example, 20 and 35 are divisible by 5.

Application of divisibility in problem solving

Divisibility principles are often used to solve mathematical problems, especially problems involving integers. Here are some examples:

Example problem 1

Determine whether the number 1872 is divisible by 9 or not.

Solution: For divisibility by 9, sum all the digits and check whether the result is divisible by 9 or not.

1 + 8 + 7 + 2 = 18

Since 18 is divisible by 9, 1872 will also be divisible by 9.

Example problem 2

Prove that if a is divisible by m and b is divisible by m, then a + b is also divisible by m.

Solution: Let a = m × k_1 and b = m × k_2 for some integers k_1 and k_2. Then:

a + b = m × k_1 + m × k_2
      = m × (k_1 + k_2)

Since k_1 + k_2 is an integer, a + b is clearly divisible by m.

Concluding remarks on divisibility

Divisibility is an important part of number theory that has wide-reaching implications in various fields of mathematics, computer science, and engineering. Understanding the properties, tests, and applications of divisibility equips us with knowledge that helps solve complex mathematical problems efficiently.

As you continue to study number theory in depth, the rules of factorization, multiples, and divisibility will be repeated constantly, proving their indispensable role in understanding the mathematical world.


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Divisibility

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