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35 LIVE: Number Theory

Ma/V 7:00–8:30pm de 9/13 (hora del Este de EE. UU.)

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El curso de LIVE tiene un total de 30 horas de formación de video en vivo (~17/hora), más 1 año de acceso a los videos pregrabados del curso del Prof. Loh.

Calendario

Las clases se darán en las fechas resaltadas. Haz clic en cualquier fecha para ver los temas del día.

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Analiza el examen asignado como tarea

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Plan de estudios

Los 20 encuentros del curso se dividen en 16 lecciones (llamadas Día 1 hasta Día 16) y 4 horas para analizar los exámenes de tarea. Cada encuentro para analizar un examen es luego de 4 lecciones.

Día 1

Remainder mod 10; definition of modular congruency; notation of a modulo b; remainders of n²; modular addition, subtraction and multiplication; remainders modulo 11; negative remainders

Día 2

Explanation and motivation for divisibility rules for 3 and 9 and shortcuts for their use; sum and average of an arithmetic progression; triangular numbers; modular multiplication

Día 3

Explanation and motivation for divisibility rules for 2, 4, and 8 and shortcuts for their use; permutations; divisibility by 12; sum and average of arithmetic progression; negative remainders

Día 4

Remainders after dividing by 99; factors; patterns in multiples of 9 mod 1; palindromic numbers; factors of 1001 and 1111; negative remainders; remainders mod 11; arithmetic progression

Día 5

Prime factorization; number of factors; sum of factors; average of factors; sum of reciprocals of factors; product of factors; factors of 111; expanding factors; sum of consecutive powers of 2; geometric series

Día 6

Number of zeroes at the end of combinatorial expressions such as factorial; modular multiplication; floor function; ways to choose n objects; sum of consecutive powers of 2;

Día 7

Least Common Multiple (LCM); Greatest Common Divisor (GCD); prime factorization; product of LCM and GCD; quotient of LCM and GCD; factorials

Día 8

Motivation for and examples of Euclidean Algorithm for finding GCD; Fibonacci numbers; factors of 111; relatively prime numbers

Día 9

Relatively prime numbers; pattern of cycling remainders; remainders of multiples of 2, 3, 4, 5, 6, 7, 8 and 9; Venn diagram; Inclusion / exclusion; Euler's Totient Function

Día 10

Chinese Remainder Theorem and use with composite moduli; negative remainders; solving sets of congruences; LCM; remainders of multiples of 6 mod 5; negative remainders

Día 11

Chinese Remainder Theorem with non-relatively-prime moduli; remainders of multiples of 9 mod 12; cycles of remainders of multiples of 9; LCM; reduction of systems of congruences; unsolvable congruences

Día 12

Systems of three congruences; Euler's Totient Function; remainders modulo composite numbers; pairwise relatively prime numbers; Venn diagram; factoring; combinatorial counting

Día 13

Factoring tricks for solving algebraic equations; area and perimeter of rectangles; number of ways to factor; equations in 1/x; impossibility of division by 0; number of integers solutions to an equation

Día 14

Remainders of powers; cycles of remainders of powers; pattern of last two digits of powers of 7; remainders of powers of 7 mod 4; power towers

Día 15

Multiplicative inverses with respect to a modulus; explanation and motivation for divisibility trick for 7; repeating cycles of remainders

Día 16

Terminating decimals and their fraction representations; repeating periods of repeating decimals; proof of why square root of 2 is irrational; prime factorization; proof techiques and directions of logic; proof by contradiction

Co-Estrellas

Foto de perfil de Vivian Loh
VIVIAN LOH

Pennsylvania MATHCOUNTS Champion • 2-time MOP qualifier • Member of US team at 2022 EGMO

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Foto de perfil de Shelly Yang
SHELLY YANG

National MATHCOUNTS competitor for Nevada • Trecretary of school math club • Enjoys video production and piano

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