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1일차
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
2일차
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
3일차
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
4일차
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
5일차
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
6일차
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;
7일차
Least Common Multiple (LCM); Greatest Common Divisor (GCD); prime factorization; product of LCM and GCD; quotient of LCM and GCD; factorials
8일차
Motivation for and examples of Euclidean Algorithm for finding GCD; Fibonacci numbers; factors of 111; relatively prime numbers
9일차
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
10일차
Chinese Remainder Theorem and use with composite moduli; negative remainders; solving sets of congruences; LCM; remainders of multiples of 6 mod 5; negative remainders
11일차
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
12일차
Systems of three congruences; Euler's Totient Function; remainders modulo composite numbers; pairwise relatively prime numbers; Venn diagram; factoring; combinatorial counting
13일차
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
14일차
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
15일차
Multiplicative inverses with respect to a modulus; explanation and motivation for divisibility trick for 7; repeating cycles of remainders
16일차
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
동료 스타
ERIC ZHAN
National MATHCOUNTS competitor for Washington • 2022 USAJMO winner • Likes to swim
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