Advanced Number Theory for Math Contests

Formalize the division theorem and residue classes modulo n. Prove the remainder of a^m−1 by a^n−1 is a^r−1 with r = m mod n, and show the order result.

Explore residue classes and modular arithmetic, proving how addition and multiplication work modulo n, and apply these ideas to powers, last-digit patterns, and contest-inspired problems.

Show that any number consisting of more than one digit of ones is not a perfect square by using modulo four, last-digit patterns, and the binomial theorem.

Explore divisibility basics, remainders, and mutual divisibility, then apply to problems involving powers, consecutive integers, and factorials to prove divisibility results and congruence ideas.

Explore divisibility techniques in number theory by proving that 25 consecutive cubes sum to a multiple of 25, using symmetry, binomial identities, and key sum formulas.

Explores number theory through previous AMC questions, proving factorial divisibility and n consecutive integers divisible by n factorial using induction and combinatorial reasoning, and introduces Euler's totient function.

Explore the pigeonhole principle and residue-class reasoning in divisibility problems. Apply these ideas to nonempty subset sums divisible by n and related Erdos-Ginzburg-Ziv ideas.

Explore the reflexive, symmetric, and transitive properties of congruence, residue classes modulo n, and how addition, subtraction, and multiplication preserve congruence, with polynomial congruences.

Explore congruences and base ten representations to prove divisibility, using constructive proofs, pigeonhole reasoning, and cyclic patterns in ten powers.

Solve prime number problems by applying modular arithmetic, divisibility arguments, and factoring tricks to prove composites and identify prime conditions in contest-style questions.

Explains Wilson's theorem and Fermat's little theorem, proves one direction of Wilson by contradiction, and presents proofs of Fermat's little theorem with applications and a preview of Chinese remainder theorem.

Apply the euclidean algorithm to gcd problems and prove gcd(a,b)=gcd(b,r). Derive x and y so gcd = a x + b y, then use Gauss's lemma on diophantine exercises.

Master the Chinese remainder theorem for solving systems of congruences with pairwise coprime moduli, using the N, y_i, z_i algorithm to obtain a unique solution modulo N.

Explore mixed number theory problems, including a sum of squares formula, modulus 1200, Chinese remainder theorem application, Fermat's little theorem exercises, and base representation challenges.

Explore AIME number theory problems, including divisors and gcd-lcm reasoning, modular arithmetic, Chinese remainder theorem, unit-digit cycles, and divisor-based problem solving strategies.

Learn factorization as a key problem solving technique in advanced number theory. The lecture demonstrates olympiad style problems where factorization yields quick solutions, including parity and square arguments.

Learn techniques to identify perfect squares and cubes using prime factorization, modular congruences, and squeezing, and apply them to olympiad problems like n^2+n+2014, x^3-5x-2, and bounds between sqrt(2n) and sqrt(5n).

Explore techniques for perfect squares and cubes through problem solving: using parity and factorization to equate squares, applying a squeezing method to cubes, and analyzing prime cases with modular reasoning.

Explores solving perfect square and cube problems in number theory contests, using modular arithmetic, parity, and difference-of-squares with case analysis for even and odd n.

Study perfect squares and cubes through problems 8–10, including two-cube cases, sums of cubes of consecutive integers, and the smallest n with a,b,c distinct giving a cube.

Explore divisors and multiplicative arithmetic functions, derive d(n) and sigma(n) from prime factorizations, and use complementary divisors and perfect square criteria in practice problems.

Explore divisors, proving results about the greatest proper divisor b of a and analyzing sums of odd and even divisors to test for perfect squares.

Prove the mean of n's divisors lies between sqrt(n) and n/2 + 1/2, and show only n = 2, 3, 6 satisfy the prime conditions for d^2 ± d + 1.

Explore divisors and modular reasoning for power-sum expressions, prove prime-like properties, and introduce base-b representations with practical digit-based problems.

Explore number bases and divisibility, solving the 15-digit pattern 7a7b7ab7b7b77 divisible by 99, and develop ABBA-type strategies to express numbers as products of distinct primes.

Explore number bases and modular reasoning through square-digit puzzles, base conversions, and contest-style problems, including finding squares ending in 2016 and interpreting polynomials with non-negative coefficients.

Explore base conversion by repeated division to convert numbers to different bases, and tackle number theory contest problems using modular reasoning, prime tests, and Sophie Germain identity.

Explore number theory problem solving with digit reversal, factorization, and cube conditions; derive the answers 132 and 165 and examine olympiad strategies and divisibility tricks.

Explore p-adic valuation and Legendre's formula to determine prime exponents in factorials, use base-p representations to compute v_p(n!), and apply these ideas to number theory problems in math contests.

Examine p-adic valuation in factorials and contest problems, from minimal factor removal to make a product end in two, to determining when n^n divides m!, using Legendre's formula.

Explore two-color grids, row and column dominance, and pigeonhole-style proofs to bound dominant squares. Examine lucky numbers in 2016x2016 tables and tiling a board with tetrominoes using coloring arguments.