MATH 3336 - Discrete Mathematics Solving Congruences (4.4 .
Page 1 of 5MATH 3336 - Discrete MathematicsSolving Congruences (4.4)Definition: A congruence of the form ππ₯ π πππ πhere m is a positive integer a andb are integers and is a variable is called a linea cong enceOur goal is to solve the linear congruence ππ₯ π πππ π that is to find all integers thatsatisf this congruenceDefini ion An integer π such that π π 1 πππ π is said to be an in e e of a modulo mExample: Show that 5 is inverse of 3 modulo 7.One method of solving linear congruences makes use of an inverse π if it e ists Althoughe cannot divide both sides of the congruence b a e can m l i l b π to solve forThe follo ing theorem guarantees that an inverse of a modulo m e ists henever a and mare relativel prime T o integers a and b are relativel prime hen πππ π, π 1Theo em If a and m are relativel prime integers and π1 then an inverse of a modulom e ists Furthermore this inverse is unique modulo m This means that there is a uniquepositive integer less than m that is an inverse of a modulo m and ever other inverse of amodulo m is congruent to modulo mE am le Find an inverse of modulo 2020, I. Perepelitsa
Page 2 of 5The Euclidean algorithm and Be out coefficients gives us a s stematic approach to findinginversesE am le Find an inverse of moduloE am le Find an inverse ofmodulo 2020, I. Perepelitsa
Page 3 of 5E am le What is the solution of the linear congruence 3π₯ 4 πππ 7The Chine e Remainde Theo emIn the first centur the Chinese mathematician Sun Tsu askedThere are certain things hose number is unkno n When divided bthe remainderishen divided bthe remainder ishen divided bthe remainder is Whatill be the number of thingsThis pu le can be translated into the solution of the s stem of congruencesπ₯ 2 πππ 3 ,π₯ 3 πππ 5 ,π₯ 2 πππ 7 .We ll see ho the theorem that is kno n as the Chine e Remainde Theo em can beused to solve Sun Tsu s problemTheorem: (The Chinese Remainder Theorem) Let π1 , π2 , , ππ be pairwise relativelyprime positive integers greater than one and π1 , π2 , , ππ arbitrary integers. Then thesystemπ₯ π1 πππ π1π₯ π2 πππ π2 π₯ π1 πππ π1has a unique solution modulo π π1 π2 ππ(That is, there is a solution π₯ with 0modulo m to this solution.)π₯π and all other solutions are congruent 2020, I. Perepelitsa
Page 4 of 5We construct a solution for congruences i e π 3 from Sun Tsu s problem Solution foran π can be constructed in a similar aπ₯ 2 πππ 3 ,π₯ 3 πππ 5 ,π₯ 2 πππ 7Step 1: π1 , π2 , π3 , π1 , π2 , π3 .Step 2: Check that π1 , π2 , π3 are pairwise relatively prime. YES or NOStep 3: Compute π1 π2 π3 , π2 π1 π3 , π3 π1 π2 , π π1 π2 π3π1 , π2 , π3 , π .Step 4: Find an inverse π¦1 of π1 modulo π1 . π¦1 .Find an inverse π¦2 of π2 modulo π2 . π¦2 .Find an inverse π¦3 of π3 modulo π3 . π¦1 .Step 5: Compute π₯ π1 π1 π¦1 π2 π2 π¦2 π3 π3 π¦3 .Step 6: Verify π₯ 2 πππ 3 , π₯ 3 πππ 5 , π₯ 2 πππ 7Try this one: Fifteen pirates steal a stack of identical gold coins. When they try to dividethem evenly, two coins are left over. A fight erupts and one of the pirates is killed. Theremaining pirates try again to evenly distribute the coins. This time there is one coin leftover. A second pirate is killed in the resulting argument. Now when the remaining piratestry to divide the coins evenly there are no coins left over. Use the Chinese RemainderTheorem to find the smallest number of coins that could have been in the sack. 2020, I. Perepelitsa
Page 5 of 5Theo em Fethen aaLimode Theo em If is prime and a is an integer not divisible bFurthermore for ever integer a e have aa modFermat s little theorem is useful in computing the remainders modulo of large po ersof integersE am le Findmod 2020, I. Perepelitsa
MATH 3336 - Discrete Mathematics Solving Congruences (4.4) Definition: A congruence of the form T : I K @ I ; Γ‘ here m is a positive integer Γ‘ a and b are integers and is a variable is called a linea congence Γ€ Our goal is to solve the linear congruence
MATH 3336 - Discrete Mathematics Recurrence Relations (8.1, 8.2) Definition: A recurrence relation { }
What is Discrete Mathematics? Discrete mathematics is the part of mathematics devoted to the study of discrete (as opposed to continuous) objects. Calculus deals with continuous objects and is not part of discrete mathematics. Examples of discrete objects: integers, distinct paths to travel from point A
CSE 1400 Applied Discrete Mathematics cross-listed with MTH 2051 Discrete Mathematics (3 credits). Topics include: positional . applications in business, engineering, mathematics, the social and physical sciences and many other ο¬elds. Students study discrete, ο¬nite and countably inο¬nite structures: logic and proofs, sets, nam- .
Calculus tends to deal more with "continuous" mathematics than "discrete" mathematics. What is the difference? Analogies may help the most. Discrete is digital; continuous is analog. Discrete is a dripping faucet; continuous is running water. Discrete math tends to deal with things that you can "list," even if the list is infinitely .
Discrete Mathematics is the part of Mathematics devoted to study of Discrete (Disinct or not connected objects ) Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous . As we know Discrete Mathematics is a back
Precalculus Any Math for College course Calc Discrete Math AP or dual - enrollment Prob & Stats Any Math for College course Discrete Math AP or dual-enrollment Math for Data and Financial Literacy (Honors) Prob Discrete Math-Algebra 2 (Honors) Precalculus Prob & Stats AP or dual It is important to note this is just a sample and other courses .
2.1 Sampling and discrete time systems 10 Discrete time systems are systems whose inputs and outputs are discrete time signals. Due to this interplay of continuous and discrete components, we can observe two discrete time systems in Figure 2, i.e., systems whose input and output are both discrete time signals.
3 PRACTICE TEST 01 May 2004 Question 1-10 All mammals feed their young. Beluga whale mothers, for example, nurse their calves for some twenty months, until they are about to give birth again and their young are able to
