# 1.3 Divisibility Rulesmr. Mac's Page

A divisibility rule is a shorthand way of discovering whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, and they are all different, we present rules only for decimal numbers. 1 Divisibility rules for numbers 120 2 Step-by-step examples 2.1 Divisibility by 2 2.2. Divisibility and Greatest Common Divisors. To install LaTeX for a Mac use MacTeX and as a front end use Texmaker.

What is P k+1?) 3 3 1. K is k is 3 3 2) Example - proving a divisibility statement is true for all positive integers n: To Prove: 3 divides n 3 + 2n when n 0. Proof: Setup: Define P n to be the statement: 3 divides n n n P 3 2 + 2200 ∈ We wish to show that P n is true for all integers n in P. Basis Step: We must show P 1 is true, that is. Divisibility rules, or divisibility tests, have a wide range of applications in mathematics (finding factors, determining prime vs. Composite, simplifying fractions, probability, etc.), but are often underemphasized in the classroom or not explored in enough detail for students to retain and use the. Free worksheet(pdf) and answer key on divisibility rules. Over 25 scaffolded questions that start relatively easy and end with some real challenges.

< Wikipedia:WikiProject Mathematics
 Main page Discussion Content Assessment Participants Resources

This is the list of all math-related draft pages in Draft as well as some in user pages (excluding redirects).

Usage

• Add pages in the draftspace that are within the scope of WikiProject Mathematics (excluding redirects). Remove them as they get moved, redirected or deleted. If necessary, note the removal in the talkpage.
• Similarly add/remove a draft page in your user page, if you wish, to let the others know about it (to avoid duplicated efforts).

• the subpages of User:Math-drafts, which is a user page used as an alternative to the draftspace.

## Concepts

### Geometry

• Draft:Faithfully flat descent - partially published to faithfully flat descent
• User:Math-drafts/Hamiltonian group action - has a broader scope than moment map
• Draft:Residual intersection - the page has been published in mainspace except one incomplete section
• Draft:Pentacontahenagon - not sure if it is notable or not.

### Algebra, algebraic topology and category theory

• Draft:Division by infinity - needs further cleanup, but could be a good counterpart to division by zero
• Draft:Eigencircle of a 2x2 matrix - it probably makes sense to have eigencircle first though.
• Draft:Frobenius formula - work out the derivation that will be put back to Frobenius formula
• Draft:Correspondence (mathematics) - should be merged with binary relation

### Mathematical analysis

• Draft:Lie's formula keep at Wikipedia:Miscellany_for_deletion/Draft:Lie's_formula_(2nd_nomination) in Feb 2018
• Draft:Bose integral - need a lot more work but the topic seems legit
• Draft:Leimkuhler-Matthews method - likely notable

### Differential equations and dynamical system

• Draft:Separable ordinary differential equation - likely covered already in mainspace

### Probability and statistics

• Draft:Vecchia approximation - a page moved from mainspace for incubation

### Number theory

• Draft:Power of 6 - barely a stub
• Draft:Gaussian symbol - probably should be part of some existing article

### Mathematical physics

• Draft:Axiomatic thermodynamics - (the question 'is thermodynamics axiomatic' seems controversial but perhaps that deserves the discussion.)
• Draft:Heaviside-Feynman formula - would be ready to go if a couple secondary sources were provided

### Combinatorics and graph theory

• Draft:Counting lemma - seems notable? it's related to Graph removal lemma (but perhaps a separate article is warranted).

### Logic and set theory

• Draft:Mathematical correspondence - should probably be merged With binary relation

### Others

• Draft:Universal approximation theorem - need to be merged with Universal approximation theorem

## Mathematicians

Put them in the alphabetical order (last name). Also, for the sake of efficiency, add only those who seem to have some potential of satisfying the notability requirement.
• Draft:Tomoyuki Arakawa - likely notable as an invited speaker at ICM
• Draft:Elena Mantovan - unclear notability
• Draft:Oscar Garcia Prada - not sufficiently notable?
• Draft:Kasia Rejzner - notability is in dispute
• Draft:Wilhelm Schlag - an invited speaker at ICM

## Uncategorised

• User:TakuyaMurata/sandbox - please do not edit it but if you find some materials useful, you’re welcome to use them in the main or draft space.

## Department Information

Graduates from the Department of Mathematics might take a job that uses their math major in an area like statistics, biomathematics, operations research, actuarial science, mathematical modeling, cryptography, or mathematics education. Or they might continue into graduate school leading to a research career. Professional schools in business, law, and medicine appreciate mathematics majors because of the analytical and problem solving skills developed in the math courses.
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#### CONTACT

Email 352.294.2350

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#### A student can receive, at most:

• Four credits for MAC 1147 and MAC 1140
• Four credits for MAC 1147 and MAC 1114
• Five credits for MAC 1140 and MAC 1114
• Five credits for MAC 1147, MAC 1140, and MAC 1114
• If both MAC 2233 and MAC 2311 (or MAC 3472) are taken, credit will be given only for MAC 2311 (or MAC 3472).

## Courses

MAA 4102 Introduction to Advanced Calculus for Engineers and Physical Scientists 1 3 Credits

Theory of real numbers, functions of one variable, sequences, limits, continuity and differentiation; continuity and differentiability of functions of several variables. Those who plan to do graduate work in mathematics should take MAA 4211. Credit will be given for, at most, one of MAA 4102, MAA 4211, or MAA 5104.

Prerequisite: (MAC 2313 or MAC 3474) and (MAS 4105 or MAS 3114) with minimum grades of C.

MAA 4103 Introduction to Advanced Calculus for Engineers and Physical Scientists 2 3 Credits

Continues the advanced calculus for engineers and physical scientists sequence. Theory of integration, transcendental functions and infinite series. MAA 4102 is not recommended for those who plan to do graduate work in mathematics; these students should take MAA 4212. Credit will be given for, at most, one of MAA 4103, MAA 4212 and MAA 5105.

Prerequisite:MAA 4102 with minimum grade of C.

MAA 4211 Advanced Calculus 1 3 Credits Advanced treatment of limits, differentiation, integration and series. Includes calculus of functions of several variables. Credit will be given for, at most, one of MAA 4211, MAA 4102 and MAA 5104.

Prerequisite:MAS 4105 with minimum grade of C.

MAA 4212 Advanced Calculus 2 3 Credits

Continues the advanced calculus sequence in limits, differentiation, integration and series. Credit will be given for, at most, one of MAA 4212, MAA 4103 and MAA 5105.

Prerequisite:MAA 4211 with minimum grade of C, taken the previous semester.

MAA 4226 Introduction to Modern Analysis 1 3 Credits

Topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables, Stokes' theorem and the Lebesgue theory. Credit will be given for, at most, MAA 4226 or MAA 5228.

Prerequisite:MAA 4212 with minimum grade of C.

MAA 4227 Introduction to Modern Analysis 2 3 Credits

Continues the modern analysis sequence discussing the topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables, Stokes' theorem and the Lebesgue theory. Credit will be given for, at most, MAA 4227 or MAA 5229.

Prerequisite:MAA 4226 with minimum grade of C, taken the previous semester.

MAA 4402 Functions of a Complex Variable 3 Credits

Complex numbers, analytic functions, Cauchy-Riemann equations, harmonic functions, elementary functions, integration, Cauchy-Goursat theorem, Cauchy integral formula, infinite series, residues and poles, conformal mapping. Credit will be given for, at most, MAA 4402 or MAA 5404.

Prerequisite: (MAC 2313 or MAC 3474) and MAP 2302 with minimum grades of C.

MAC 1105 Basic College Algebra 3 Credits

Online entry-level algebra course for college students. (M)

Prerequisite: completion of the ALEKS placement exam.

Attributes: General Education - Mathematics

MAC 1114 Trigonometry 2 Credits

Exponential and logarithmic functions, trigonometry and analytic and additional applications of trigonometry. (M)

Attributes: General Education - Mathematics

MAC 1140 Precalculus Algebra 3 Credits

College algebra, functions, coordinate geometry, exponential and logarithmic functions. (M)

Prerequisite: completion of the ALEKS placement exam.

Attributes: General Education - Mathematics

MAC 1147 Precalculus Algebra and Trigonometry 4 Credits

College algebra, functions, coordinate geometry, exponential and logarithmic functions, and trigonometry. Fast-paced review of algebra and trigonometry to prepare for calculus. Assumes prior knowledge of intermediate algebra (Algebra 2) and trigonometry. (M)

Prerequisite: at least 50% on the ALEKS placement exam.

Attributes: General Education - Mathematics

MAC 2233 Survey of Calculus 1 3 Credits

Geometric and heuristic approach to calculus; differentiation and integration of simple algebraic and exponential functions; applications to graphing, marginal analysis, optimization, areas and volumes. (M)

Prerequisite: Any of the following: minimal acceptable score on the online mathematics placement exam; a minimum grade of C in a MAC course numbered 1140 or higher; AP credit for MAC 2311; IB credit for a MAC course numbered 1140 or higher. Any course grades, AP or IB scores used to meet this prerequisite must be on file at UF by registration.

Attributes: General Education - Mathematics

MAC 2234 Survey of Calculus 2 3 Credits

Sequences, geometric and Taylor series; systems of linear equations, Gaussian elimination, matrices, determinants and vectors; partial differentiation, multiple integrals; applications to marginal analysis, least-squares and Lagrange multipliers. (M)

Prerequisite:MAC 2233 with minimum grade of C, or the equivalent.

Attributes: General Education - Mathematics

MAC 2311 Analytic Geometry and Calculus 1 4 Credits

Introduces analytic geometry; limits; continuity; differentiation of algebraic, trigonometric, exponential and logarithmic functions; applications of the derivative; inverse trigonometric functions; differentials; introduction to integration; and the fundamental theorem of calculus. (M) Credit will be given for, at most, one of MAC 2233, MAC 2311 and MAC 3472.

Prerequisite: Any of the following: minimal acceptable score on the online mathematics placement exam; a grade of C in a MAC course numbered 1147 or higher; AP credit for MAC 2311; IB credit for a MAC course numbered 1147 or higher. Any course grades, AP or IB scores used to meet this prerequisite must be on file at UF by registration.

Attributes: General Education - Mathematics

MAC 2312 Analytic Geometry and Calculus 2 4 Credits

Techniques of integration; applications of integration; differentiation and integration of inverse trigonometric, exponential and logarithmic functions; sequences and series. (M) Credit will be given for, at most, one of MAC 2312, MAC 2512 and MAC 3473.

Prerequisite:MAC 2311 or MAC 3472 with a minimum grade of C.

Attributes: General Education - Mathematics

MAC 2313 Analytic Geometry and Calculus 3 4 Credits

Solid analytic geometry, vectors, partial derivatives and multiple integrals. (M) Credit will be given for, at most, MAC 2313 or MAC 3474.

Prerequisite:MAC 2312 or MAC 2512 or MAC 3473 with a minimum grade of C.

Attributes: General Education - Mathematics

MAC 2512 Calculus 2 for Advanced Placement Students 4 Credits

For entering freshmen who have Advanced Placement Calculus AB credit for MAC 2311. MAC 2512 covers those topics in MAC 2311 and MAC 2312, which is not included or only partially covered in the AP Calculus AB curriculum. Some topics from the AP curriculum are reviewed briefly in the first part of the semester. The combination of AP Calculus AB and MAC 2512 has the same content as the MAC 2311/2312 sequence. Calculus 2 topics to which the student has been exposed in AP Calculus AB are covered more quickly in MAC 2512 than in MAC 2312. (M) Credit will be given for, at most, one of MAC 2312, MAC 2512 and MAC 3473.

Prerequisite: AP credit for MAC 2311.

Attributes: General Education - Mathematics

MAC 3472 Honors Calculus 1 4 Credits

Topics covered in the MAC 3472/MAC 3473/MAC 3474 sequence closely parallel those covered in MAC 2311/MAC 2312/MAC 2313, but are treated in greater depth. Credit will be given for, at most, MAC 2311 or MAC 3472. (M)

Prerequisite: strong background in precalculus.

Attributes: General Education - Mathematics

MAC 3473 Honors Calculus 2 4 Credits

Continues the honors calculus sequence. (M) Credit will be given for, at most, one of MAC 2312, MAC 2512 and MAC 3473.

Prerequisite:MAC 3472 or MAC 2311 with a minimum grade of C.

Attributes: General Education - Mathematics

MAC 3474 Honors Calculus 3 4 Credits

Continues the honors calculus sequence. (M) Credit will be given for, at most, MAC 2313 or MAC 3474.

Prerequisite:MAC 2312 or MAC 2512 or MAC 3473 with a minimum grade of C.

Attributes: General Education - Mathematics

MAD 2502 Intro to Computational Math 3 Credits

### 1.3 Divisibility Rulesmr. Mac's Page Key

Is an introduction to mathematical computation and the Python programming language. Emphasis is on using mathematical algorithms to solve problems in analysis, number theory, combinatorics, algebra, linear algebra, numerical analysis, and probability.

Prerequisite:MAC 2311 or MAC 3472, minimum grade of C.

MAD 3107 Discrete Mathematics 3 Credits

Logic, sets, functions; algorithms and complexity; integers and algorithms; mathematical reasoning and induction; counting principles; permutations and combinations; discrete probability. Advanced counting techniques and inclusion-exclusion.

Prerequisite:MAC 2312 or MAC 2512 or MAC 3473 with a minimum grade of C.

MAD 4203 Introduction to Combinatorics 1 3 Credits

Permutations and combinations, binomial coefficients, inclusion-exclusion, recurrence relations, Fibonacci sequences, generating functions and graph theory.

Prerequisite: (MAC 2312 or MAC 2512 or MAC 3473) and (MAS 3300 or MHF 3202) with minimum grades of C.

MAD 4204 Introduction to Combinatorics 2 3 Credits

Matching theory, block designs, finite projective planes and error-correcting codes. Does not require MAD 4203.

Prerequisite: (MAC 2312 or MAC 2512 or MAC 3473) and (MAS 3300 or MHF 3202) with minimum grades of C.

MAD 4401 Introduction to Numerical Analysis 3 Credits

Numerical integration, nonlinear equations, linear and nonlinear systems of equations, differential equations and interpolation.

Prerequisite:MAS 3114 or MAS 4105 with a minimum grade of C and experience with a scientific programming language.

MAE 3811 Mathematics for Elementary School Teachers 2 3 Credits

Properties of and operations with rational numbers; ratio; proportion; percentages; an introduction to real numbers; elementary algebra; informal geometry and measurement; and introduces probability and descriptive statistics.

Prerequisite: College of Education major.

MAP 2302 Elementary Differential Equations 3 Credits

First-order ordinary differential equations, theory of linear ordinary differential equations, solution of linear ordinary differential equations with constant coefficients, the Laplace transform and its application to solving linear ordinary differential equations. (M)

### 1.3 Divisibility Rulesmr. Mac's Page Printable

Prerequisite:MAC 2312 or MAC 2512 or MAC 3473 with a minimum grade of C.

Attributes: General Education - Mathematics

MAP 2483 Mathematical Methods for Natural Sciences 4 Credits

Introduces basic mathematical methods and computer modeling used in the natural sciences, including data representation and analysis, basic statistics and probability, linear algebra, stochastic and deterministic processes and optimization. Theoretical concepts are integrated with real-life applications and computer modeling projects.

Prerequisite:MAC 2311.

MAP 4102 Probability Theory and Stochastic Processes 2 3 Credits

Random walks and Poisson processes, martingales, Markov chains, Brownian motion, stochastic integrals and Ito's formula.

Prerequisite:STA 4321 with a minimum grade of C.

MAP 4305 Differential Equations for Engineers and Physical Scientists 3 Credits

### 1.3 Divisibility Rulesmr. Mac's Page Sheet

The second course in differential equations. Topics include systems of linear differential equations, stability theory and phase plane analysis, power series solutions of differential equations, Sturm-Liouville boundary-value problems and special functions. Credit will be given for, at most, MAP 4305 or MAP 5304.

Prerequisite:MAP 2302 and (MAS 3114 or MAS 4105 or EGM 3344) with minimum grades of C.

MAP 4341 Elements of Partial Differential Equations 3 Credits

Introduces second-order linear partial differential equations (heat, wave and Laplace equations), separation of variables in PDEs, Sturm-Liouville eigenvalue problems, method of eigenfunction expansions (Fourier analysis) and Green's functions. Possible introduction to first-order PDEs and the method of characteristics. Credit will be given for, at most, MAP 4341 or MAP 5345.

Prerequisite:MAP 2302 and MAP 4305 with minimum grades of C.

MAP 4413 Fourier Analysis 3 Credits

Introduces linear systems and transforms; Laplace, Fourier and Z transforms and their mutual relationship; convolutions. Operational calculus; computational methods including the fast Fourier transform; second-order stationary processes and their autocorrelation functions; and problems of interpolation, extrapolation, filtering and smoothing of second-order stationary processes.

Prerequisite: (MAC 2313 or MAC 3474) and MAP 2302 and (MAS 3114 or MAS 4105) with minimum grades of C.

MAP 4484 Modeling in Mathematical Biology 3 Credits

Mathematical models of biological systems. Topics include models of growth, predator-prey populations, competition, the chemostat, epidemics, excitable systems and analytical tools such as linearization, phase-plane analysis, Poincare-Bendixson theory, Lyapunov functions and bifurcation analysis.

Prerequisite:MAP 2302 and (MAS 3114 or MAS 4105) with minimum grades of C.

MAS 3114 Computational Linear Algebra 3 Credits

Linear equations, matrices and determinants. Vector spaces and linear transformations. Inner products and eigenvalues. Emphasizes computational aspects of linear algebra.

Prerequisite:MAC 2312, MAC 2512 or MAC 3473 with a minimum grade of C and experience with a scientific programming language.

MAS 3300 Numbers and Polynomials 3 Credits

Emphasizes theorems and proofs. Topics include algebraic and order properties of the real numbers; introduction to number theory; rational numbers and their decimal expansions; uncountability of the real numbers; complex numbers, irreducible polynomials over the integral, rational, real and complex numbers; and elementary theory of equations. Taking one, but not both, of MAS 3300 or MHF 3202 is required of mathematics majors. MAS 3300 is also particularly useful for prospective secondary-school mathematics teachers. (M)

Prerequisite: a UF math course at the 2000 level or above with a minimum grade of C; this requirement is waived for transfer students with junior standing.

Attributes: General Education - Mathematics

MAS 4105 Linear Algebra 1 4 Credits

Linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues and inner-product spaces. Includes both theory and computational skills. Develops the ability to reason through, and coherently write, proofs of theorems. For math majors, this course serves as a transition from a study of techniques into more conceptual math; for engineering and science majors, it serves also as a coherent foundation in linear algebra.

Prerequisite: (MAC 2313 or MAC 3474) and (MAS 3300 or MHF 3202) with minimum grades of C.

MAS 4124 Introduction to Numerical Linear Algebra 3 Credits

Topics in linear algebra most useful in applications with emphasis on the numerical methods involved: direct and iterative solutions to systems of linear equations; matrix norms; Householder transformations; singular value decomposition; least squares and the generalized inverse; QR method for computing eigenvalues; condition number of linear systems and eigensystems.

Prerequisite:MAS 3114 or MAS 4105 with a minimum grade of C and experience with a scientific programming language.

MAS 4203 Introduction to Number Theory 3 Credits

Introduces elementary number theory and its applications to computer science and cryptology. Divisibility, primes, Euclidean Algorithm, congruences, Chinese Remainder Theorem, Euler-Fermat Theorem and primitive roots. Selected applications to decimal fractions, continued fractions, computer file storage and hashing functions, and public-key cryptography.

Prerequisite:MAC 2312 and (MAC 2512 or MAC 3473) with a minimum grade of C; MAS 3300 recommended.

MAS 4301 Abstract Algebra 1 3 Credits

Sets and mappings, groups and subgroups, homomorphisms and isomorphisms, permutations, rings and domains, arithmetic properties of domains, and fields. Requires facility in writing proofs.

Prerequisite: (MAS 3300 or MHF 3202 with a minimum grade of B) or MAS 4105 with a minimum grade of C.

MAT 3503 Functions and Modeling 3 Credits

Group activities strengthen knowledge of secondary mathematics, especially topics from precalculus and the transition to calculus, including contexts that can be modeled using linear, exponential, polynomial or trigonometric functions. Topics include conic sections, parametric equations and polar equations. Explorations involve multiple representations, transformations and data analysis techniques, and are facilitated by various technologies.

Prerequisite:MAC 2311 and UFTeach Step 1.

Corequisite:MAC 2312.

MAT 4905 Individual Work 1-3 Credits

Special topics not obtainable in regular course offerings.

Prerequisite:MAC 2313 or MAC 3474 with a minimum grade of C and undergraduate coordinator permission.

MAT 4911 Undergraduate Research in Mathematics 0-3 Credits

Provides firsthand, supervised research in mathematics. Projects may involve inquiry, design, investigation, scholarship, discovery or application in mathematics.

MAT 4930 Special Topics in Mathematics 1-3 Credits

Qualified undergraduates take part in seminars or classes on special topics.

MAT 4956 Overseas Studies 1-15 Credits  Provides a mechanism by which coursework taken as part of an approved study abroad program can be recorded on the UF transcript and counted toward graduation.

MGF 1106 Mathematics for Liberal Arts Majors 1 3 Credits

For non-science and non-business majors who need to fulfill the writing and general education math requirements. Includes an introduction to set theory, logic, number theory, probability, statistics, graphing and linear programming. (M)

Attributes: General Education - Mathematics

### 1.3 Divisibility Rulesmr. Mac's Page Shortcut MGF 1107 Mathematics for Liberal Arts Majors 2 3 Credits

General-education course that demonstrates the beauty and utility of mathematics. Topics include financial management, linear and exponential growth, mathematics in the arts and discrete mathematics. Does not require MGF 1106. (M)

Attributes: General Education - Mathematics

MHF 3202 Sets and Logic 3 Credits

Examples of sets, operations on sets, set algebra, Venn diagrams, truth tables, tautologies, applications to mathematical arguments and mathematical induction. Taking one, but not both, of MAS 3300 or MHF 3202 is required of mathematics majors. MHF 3202 can also be very useful for prospective and in-service secondary and middle school teachers. (M)

Prerequisite: a UF math course at the 2000 level or above with a minimum grade of C.

Attributes: General Education - Mathematics

MHF 4102 Elements of Set Theory 3 Credits

Basic axioms and concepts of set theory. Students present proofs. Credit will be given for, at most, MHF 4102 or MHF 5107.

Prerequisite:MAS 4105 with a minimum grade of C.

MHF 4203 Foundations of Mathematics 3 Credits

Models and proofs. Foundations of real and natural numbers, algorithms, Turing machines, undecidability and independence. Examples and applications in algebra, analysis, geometry and topology. Credit will be given for, at most, MHF 4203 or MHF 5207.

Prerequisite:MAS 4105 with a minimum grade of C.

MTG 3212 Geometry 3 Credits

Axiomatic treatment of topics in Euclidean, non-Euclidean, projective geometry and (time permitting) fractal geometry. Particularly useful for prospective secondary-school mathematics teachers.

Prerequisite:MAC 2312 and (MAC 2512 or MAC 3473 with a minimum grade of C).

MTG 3214 Euclidean Geometry 3 Credits

Axiomatic structure of Euclidean geometry: congruence, parallelism, area, similarity, circles, polygons, medians, constructions, solid geometry, spherical and hyperbolic geometry. Particularly useful for prospective secondary-school mathematics teachers.

Prerequisite:MAC 2312 and (MAC 2512 or MAC 3473 with a minimum grade of C).

MTG 4302 Elements of Topology 1 3 Credits