PDF4PRO ⚡AMP

Modern search engine that looking for books and documents around the web

Example: biology

BASIC PROPERTIES OF CONGRUENCES

BASIC PROPERTIES OF CONGRUENCESThe lettersa, b, c, d, krepresent integers. The lettersm, nrepresent positive integers. Thenotationa b(modm) means thatmdividesa b. We then say thatais congruent (Reflexive Property):a a(modm)2. (Symmetric Property): Ifa b(modm), thenb a(modm).3. (Transitive Property): Ifa b(modm) andb c(modm), thena c(modm).Remark:The above three PROPERTIES imply that (modm) is an equivalence relationon the Ifa b(modm) andc d(modm), thena+c b+d(modm) anda c b d(modm).5. Ifa b(modm) andc d(modm), thenac bd(modm).6. Assume thata b(modm). Letk 1. Thenak bk(modm).7. Suppose thatP(x) is any polynomial with coefficients inZ. Assume thata b(modm).ThenP(a) P(b) (modm).8. Assume thata b(modm). Thengcd(a, m) =gcd(b, m).9. Ifa b(modm) andn|m, thena b(modn).

BASIC PROPERTIES OF CONGRUENCES The letters a;b;c;d;k represent integers. The letters m;n represent positive integers. The notation a b (mod m) means that m divides a b. We then say that a is congruent to b modulo m. 1. (Re exive Property): a a (mod m) 2. (Symmetric Property): If a b (mod m), then b a (mod m). 3.

Fullscreen Download
Loading..

Tags:

  Properties, Properties of

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Spam in document Broken preview Other abuse

Transcription of BASIC PROPERTIES OF CONGRUENCES

Documents from same domain

THE COMPLEX EXPONENTIAL FUNCTION

sites.math.washington.edu

THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following de nition ... we will rst verify the following form of the Law of Exponents: ei 1+i 2 = ei 1 ei 2 (2) To prove this we rst expand the right-hand side of (1) by rst multiplying out the

  Functions, Properties, Complex, Exponential, Exponent, Of exponents, The complex exponential function

Working a difference quotient involving a square root

sites.math.washington.edu

Working a difference quotient involving a square root Suppose f(x) = p x and suppose we want to simplify the differnce quotient f(x+h) f(x) h as much as possible (say, to eliminate the h …

  Square, Working, Differences, Involving, Quotient, Working a difference quotient involving a square

GALOIS THEORY - University of Washington

sites.math.washington.edu

GALOIS THEORY We will assume on this handout that is an algebraically closed eld. This means that every irreducible polynomial in [x] is of degree 1. Suppose that F is a sub eld of and that Kis a nite extension of Fcontained in . For example, we can take = C, the eld

  Theory, Galois theory, Galois

Galois Field in Cryptography - sites.math.washington.edu

sites.math.washington.edu

Galois Field in Cryptography Christoforus Juan Benvenuto May 31, 2012 Abstract This paper introduces the basics of Galois Field as well as its im-

  Field, Cryptography, Galois, Galois field in cryptography

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

sites.math.washington.edu

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [a;b] be a bounded interval in R. C2([a;b]) denotes the space of functions with derivatives of second order continuous up to

  Value, Problem, Boundary, Sturm, Liouville, Sturm liouville boundary value problems

Markov Chains - University of Washington

sites.math.washington.edu

924 CHAPTER17 Markov Chains ter the coin has been flipped for the tth time and the chosen ball has been painted.The state at any time may be described by the vector [urb], where uis the number of un-painted balls in the urn, is the number of red balls in the urn, and r …

  Chain, Markov, Markov chain

ZZ dA R [0 2] R - University of Washington

sites.math.washington.edu

Math 126C 2nd Midterm Solutions Spring 2014 3 (8points) Computetheequationofthetangentlinetothecurve r =1+2sinθ atthe pointwhereθ =π/6. Giveyouranswerinexactform.

  D ra

Precalculus - University of Washington

sites.math.washington.edu

Precalculus David H. Collingwood Department of Mathematics University of Washington K. David Prince Minority Science and Engineering Program College of Engineering

  Precalculus

Precalculus - University of Washington

sites.math.washington.edu

iv of study per week (outside class) is a more typical estimate. In other words, for many students, this course is the equivalent of a half-time job!

  Precalculus

Electrical Circuits - University of Washington

sites.math.washington.edu

Linear Algebra in Electrical Circuits Perhaps one of the most apparent uses of linear algebra is that which is used in Electrical Engineering. As most students of mathematics have encountered, when the

  Electrical, Circuit, Electrical circuits

Related documents

Structure and Mechanical Properties of Materials

sig.ias.edu

and properties of materials •A simple introduction to amorphous and crystalline structure was presented •This was followed by some basic definitions of stress, strain & mechanical properties •The mechanical properties of soft and hard tissue were then introduced •Balance of mechanical properties is key for design

  Properties, Properties of

Lecture 5: Homogeneous Equations and Properties of Matrices

dkatz.ku.edu

Lecture 5: Homogeneous Equations and Properties of Matrices Lecture 5: Homogeneous Equations and Properties of Matrices. De nition A system of linear equations is said to be homogeneous if the right hand side of each equation is zero, i.e., each equation in the system has the form a 1x 1 + a 2x 2 + + a nx n = 0:

  Properties, Properties of

Chemical & Physical Properties of Crude Oil

osha.washington.edu

Characteristics of Crude Oil •The hydrocarbons in crude oil can generally be divided into four categories: •Paraffins: These can make up 15 to 60% of crude. •Paraffins are the desired content in crude and what are used to make fuels. •The shorter the paraffins are, the lighter the crude is.

  Properties, Properties of

Probability, Random Processes, and Ergodic Properties

ee.stanford.edu

Ergodic properties and theorems We develop the notion of time averages along with that of probabilistic averages to emphasize their similarity and to demonstrate many of the implica-tions of the existence of limiting sample averages. We prove the ergodic theorem theorem for the general case of asymptotically mean stationary processes.

  Properties, Ergodic, Ergodic properties

GROUP PROPERTIES AND GROUP ISOMORPHISM

math.ucsd.edu

May 25, 2001 · GROUP PROPERTIES AND GROUP ISOMORPHISM groups, developed a systematic classification theory for groups of prime-power order. He agreed that the most important number associated with the group after the order, is the class of the group.In the book Abstract Algebra 2nd Edition (page 167), the authors [9] discussed how to find all the abelian …

  Properties, Isomorphism

Properties of Exponents - cdn.kutasoftware.com

cdn.kutasoftware.com

Properties of Exponents Date_____ Period____ Simplify. Your answer should contain only positive exponents. 1) 2 m2 ⋅ 2m3 4m5 2) m4 ⋅ 2m−3 2m 3) 4r−3 ⋅ 2r2 8 r 4) 4n4 ⋅ 2n−3 8n 5) 2k4 ⋅ 4k 8k5 6) 2x3 y−3 ⋅ 2x−1 y3 4x2 7) 2y2 ⋅ 3x 6y2x 8) 4v3 ⋅ vu2 4v4u2 9) 4a3b2 ⋅ 3a−4b−3 12 ab 10) x2 y−4 ⋅ x3 y2 x5 y2 11) (x2 ...

  Properties, Properties of

Properties of Parabolas - cdn.kutasoftware.com

cdn.kutasoftware.com

Properties of Parabolas Date_____ Period____ Identify the vertex of each. 1) y = x2 + 16 x + 64 2) y = 2x2 − 4x − 2 3) y = −x2 + 18 x − 75 4) y = −3x2 + 12 x − 10 Graph each equation. 5) y = x2 − 2x − 3 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 6) …

  Properties, Properties of

Properties of Logarithms - Shoreline Community College

www.shoreline.edu

PROPERTIES OF LOGARITHMIC FUNCTIONS EXPONENTIAL FUNCTIONS An exponential function is a function of the form f (x)=bx, where b > 0 and x is any real number. (Note that f (x)=x2 is NOT an exponential function.) LOGARITHMIC FUNCTIONS log b x =y means that x =by where x >0, b >0, b ≠1 Think: Raise b to the power of y to obtain x. y is the exponent.

  Community, College, Properties, Shorelines, Properties of, Shoreline community college

Related search queries

PROPERTIES OF, PROPERTIES, Ergodic properties, Ergodic, ISOMORPHISM, Shoreline Community College