Discrete Mathematics 2

Explore why combinatorics matters in discrete mathematics, from counting permutations, variations, and combinations to derangements, partitions, and Stirling numbers, with links to discrete probability and algorithm analysis.

Explore how combinatorics offers multiple paths to the same destination, covering binomial coefficients, Pascal's triangle, induction, variations and combinations, and lattice-path counting.

Trace how Pascal's triangle yields n choose k through lattice paths and bit strings, and relate subsets, pizza toppings, and the binomial theorem to combinations and permutations.

Master essential terminology for coins, dice, cards, and balls in combinatorics and discrete probability, including coin outcomes, die orientation, card suits and face cards, and sticks-and-stones or stars-and-bars methods.

Explore fundamental terminology in probability and combinatorics: experiments, outcomes, events, and favorable outcomes, and learn to model them with sample space, sets, and Cartesian products.

Compare sampling with and without replacement for p and q from 1 to 10, using the graph of q=2p to identify outcomes and count favorable cases.

Explore how to combine outcomes in discrete mathematics 2 using the sum principle, product principle, and the principle of inclusion-exclusion, with examples and distinctions between disjoint and overlapping cases.

Explore the sum principle through concrete disjoint events, counting outcomes with binomial coefficients, including two-letter words and 10-bit strings of weight 6, using Pascal's triangle.

Apply the product principle to count outcomes across two or more events, reflecting cartesian product reasoning. Explore examples like two-letter words, license plates, and three-letter words with repeats and without.

Combine the sum and product principles by splitting into disjoint cases to count outcomes. Apply to card selections, lattice paths, and two-digit numbers with even digit sums.

Explore the inclusion-exclusion principle for combining overlapping outcomes, apply it to two- and three-set unions, and examine examples like weight-4 7-bit strings and multiples of 2, 3, or 5.

Explore how to compute binomial coefficients from Pascal's triangle and derive the closed formula for n choose k through factorials and variations without repetition, as covered in section 3.4.

Examine functions from a finite set K to a finite set N, counting all functions, injections, surjections, and bijections, and analyze increasing and non-decreasing maps under chosen orders.

Count triangles with vertices from 11 points arranged on perpendicular lines, avoiding degenerate cases. Explore three methods: product and sum principles, subsets, and inclusion-exclusion, arriving at 120 triangles.

Counting problem 2 in discrete mathematics 2 uses two methods: product principle and n choose 2 minus sides to count pairs of n people and diagonals in a convex polygon.

Learn to count permutations with adjacency constraints using block methods, sum principle, and bijections. See solutions for the one to 2025 set, including one and two next to each other.

Solve discrete math counting problems with 8-digit numbers from 1–9, including those divisible by 15, and extend to general n-digit numbers using De Morgan's law and inclusion–exclusion.

Explore a classical committee counting problem to prove binomial-coefficient formulas using two methods: choose chair and secretary first, then five members, or select seven and assign roles.

Explore counting four-element subsets from 1 to 3n using injections and modulo three analysis to show one third have sums divisible by three in discrete mathematics 2.

Explore counting five rooks on an eight by eight chessboard, comparing indistinguishable versus distinguishable rooks, and count non-attacking placements using column choices, row allocations, and factorials.

Explore counting techniques in poker by calculating five-card hands and full houses using combinations, probability, and sum and product rules, including incomplete decks.

Apply the product principle to prime factorization, using the cartesian product of exponent choices to count divisors; for 60 these yield 12; in general use the product of (αi+1).

Count six-digit numbers formed from distinct digits from A, determine the total permutations (6!), and count those divisible by 4 using the last-two-digits rule, yielding 192.

Count pairs p and q from 1 to 10 for which x^2 + px + q has real roots; with replacement 62 of 100 outcomes, without replacement 55 of 90.

Learn to distribute indistinguishable balls into numbered boxes and identical cookies to distinct children using a unified counting method, with zeros and ones and multisets for remainder classes.

Distribute indistinguishable objects into k boxes with no empty box, using the boxes-and-objects model and a zeros-and-ones transformation to count the positive integer solutions of x1+...+xk=n.

Explore multisets and their links to sticks and stones, balls in bins, and stars and bars, and see how sets differ from multisets through concrete examples.

Apply sticks and stones to model integer solutions, distribute identical balls into boxes with or without restrictions, and explore non-increasing seven-digit numbers and a bijection between subsets and multisets.

Explore partitions of sets, multinomial coefficients, and permutations of multisets through concrete examples, including ordered and unordered partitions, and the multinomial theorem.

Explore optional material on partitions and distributions, detailing labeled and unlabeled balls into boxes, A and X notation, and the link to the 12-fold way in enumerative combinatorics.

Explore partitions of labeled objects into nonempty unlabeled parts and learn Stirling numbers of the second kind, their recurrence, and connections to multinomial coefficients and surjections.

The twelvefold way presents twelve placements of balls into boxes under labeled or unlabeled sets and unrestricted, max-one, or no-empty conditions.

Generalize the inclusion-exclusion principle for counting unions of n finite sets. Learn and prove the generalized formula using S1, S2, S3 and beyond.

Apply the negative form of the inclusion-exclusion principle using De Morgan’s laws, pi principle, and floor function to count numbers not divisible by 3, 5, or 7 up to 1000.

Apply pi principle to multisets and sticks-and-stones to solve two examples: distributing 11 cookies with at most four per child and counting nonnegative solutions with each variable at most three.

Count surjections between finite sets using inclusion-exclusion to derive a closed formula, illustrate with six-to-four mappings, and verify via Stirling numbers.

Explore fixed points and derangements of permutations, counting permutations with no fixed points using the inclusion-exclusion principle, with an illustrative 4-element example yielding 9 derangements.

Explore the absent-minded secretary problem in discrete mathematics 2, analyze permutations and derangements, and apply counting and probability methods to determine cases with at least one fixed point.

Explore combinatorial proofs and binomial identities in discrete mathematics 2, examining binomial coefficients, direct proofs, induction, and telescoping sums, and locate key formulas and derivations on the formula sheet.

Explore a delightful non-binomial sum problem that uses Gauss's formulas for sums of the first natural numbers, their squares, and cubes, with an index change and odd-number denominators.

Explore methods for proving formulas, including direct proofs, induction, telescoping sums, and combinatorial proofs, using binomial identities, Pascal's triangle, and the binomial theorem, with the formula sheet.

Explore telescoping sums with factorials and prove a formula by induction using the recursive factorial definition. See base case and induction step illustrated.

Demonstrate a factorial telescoping sum formula for all n≥1 using induction and telescoping techniques, with base case and induction step.

Explore the absorption-extraction formula for binomial coefficients, proving n choose k equals n/k times (n-1) choose k-1 with direct and combinatorial proofs. Illustrate via a committee with a chairman.

Demonstrate a direct proof of a trinomial identity, swapping m and k for alignment, and analyze multinomial and quadrinomial coefficients through combinatorial examples like Mississippi.

Explore the hockey stick identity for binomial coefficients via diagonal summation, using induction, telescoping sums, and a combinatorial proof in Pascal's triangle, with a lazy twinship approach.

Explore the parallel summation hockey stick identity for binomial coefficients and its symmetry with diagonal summation, using Pascal's triangle to show equal sums and a lazy, concise proof.

Explore the sum of squares identity for binomial coefficients, with three proofs—two combinatorial (pizza toppings and lattice paths) and one via the binomial theorem, plus a primer on polynomials.

Explore how to generate many binomial formulas by applying the binomial theorem with different a and b values, deriving identities for 2^n, 3^n, 4^n, and beyond.

Present an alternate view of the diagonal summation formula and prove its validity with the binomial theorem and coefficient interpretation, linking the left-hand side to coefficients of 1+x powers.

Rewrite a clean diagonal summation solution using binomial theorem and sigma notation for n=80, m=2, and verify the coefficient of x squared equals n+1 choose m+1, confirming video 55.

Explore two elegant combinatorial proofs of 3^n = sum_{k=0}^n 2^k C(n,k). Count ternary digit strings and pairs of sets A ⊆ B ⊆ [n] using encodings.

Learn a combinatorial proof counting 3-element subsets of an n+2 set in two ways, then generalize to 5-element subsets of an n+3 set using the middle element method.

Explore Vandermonde identity through a combinatorial proof of choosing a team from n boys and m girls, equating left and right sides via product and sum principles.

Explore three proofs of the identity sum_{k=1}^n k C(n,k) = n 2^{n-1}, using the absorption-extraction formula, calculus, and a combinatorial committee proof counted in two ways.

Explore proving a challenging binomial coefficient identity by applying the trinomial revision formula and Vandermonde identity, with a careful change of index and strategic notation.

Explore more binomial identities and absorption in problem 17, deriving the sum equals (2^(n+1)-1)/(n+1) using binomial coefficients, plus an optional integral approach.

Prove a binomial identity (problem 18) by rewriting with sigma notation and applying the absorption-extraction formula to relate n choose k to n+1 choose k+1, then simplify two sums.

Explore the monotonicity of binomial coefficients by proving n choose k < n choose k+1 for k < floor(n/2) using factorial cancellations and Pascal's triangle symmetry.

Explore a very brief introduction to discrete probability with finite sample spaces, not a formal course, focusing on applying combinatorial methods to compute event probabilities.

Explore how to model experiments with finite sample spaces, define events and favorable outcomes, and visualize probabilities using die-roll and coin-toss examples, including binary strings and weight concepts.

Explore how set theory underpins probability by treating sample spaces and events as sets, and learn to describe unions, intersections, and complements with practical examples.

Explore probability foundations by modeling events as sets within a sample space, using Venn diagrams for A, B, and C, and practicing set operations such as union, intersection, and complement.

Explore Kolmogorov's axiomatic definition of probability and its three axioms for events, sample spaces, and probabilities. See how additivity applies to mutually exclusive events in finite spaces.

Derive key properties of probabilities from the axiomatic definition using set theory, including additivity, monotonicity, complements, and inclusion-exclusion.

Derives the finite sample space and uniform distribution, showing probability equals favorable outcomes over the total. Illustrates additivity and union–intersection rules with combinatorial counting and dice examples.

Compute probabilities for events in two experiments—rolling a die twice and flipping a coin four times—by counting outcomes, using sample space cardinalities, and applying binomial coefficients.

Compute probabilities for two and four numbers drawn from A = {1,...,3n} with sums divisible by 3, by comparing the event to the sample space, resulting in a 1/3 probability.

Compute the probability that the roots of x^2 + p x + q are real, with p and q from 1 to 10, in cases with replacement and without replacement.

Compute probabilities for events B and C, using eight-digit numbers with distinct digits from A divisible by 15, and n-digit numbers from A whose product is divisible by 10.

This lecture computes probability that no compartment is empty when distributing n passengers into k compartments, noting zero if n<k, and when n=k indistinguishable passengers yield 1 over binomial(2n-1, n-1).

Solve a seating problem with two girls and three boys by applying combinatorics and probability, using permutations, the sample space, and complementary events to check if the girls sit together.

Apply the sticks and stones method to model nine passengers as binary strings of length eleven with two ones, and compute sample space and three probabilities 7/55, 1/55, and 6/55.

Explore how hockey stick identity provides a combinatorial proof and probability insight for counting numbers with non-decreasing digits up to 50 billion, using sticks and stones and binomial coefficients.

Explore counting pairings of 20 people to form 10 unordered pairs. Compute the probability that every pair is mixed gender using two cardinality methods and the sample space.

Explore probability in discrete mathematics 2: selecting 2R shoes from m pairs with no pair, exactly one pair, or two pairs, via the pigeonhole principle and binomial identities.

Compute poker hand probabilities using the product principle and complementary counting, covering royal flush, straight flush, four of a kind, full house, flush, straight, and high card.

This discrete mathematics lecture analyzes weird poker with a 46-card deck missing all hearts, using sum and product rules to compare full-house probabilities.

Apply the sum and product rules to probability diagrams and sample spaces, comparing with and without replacement, and illustrating events like two black, two white, and mixed colors.

Explore repeated experiments and Cartesian products of multiple sets, using a four-coin toss to compute events A and B and their intersection, illustrating sample spaces and cardinalities.

Explore probability with examples from the discrete mathematics book, using complementary events to find at least one head in ten coin tosses and a six in four die rolls.

Explore independent and dependent events through dice and coin experiments, showing when P(A∩B)=P(A)P(B) and illustrating why some event pairs are not independent.

Explore conditional probability through a practical example, computing P(A|B) for a woman who does not play football, and relate it to independent versus dependent events and the product rule.

Explore random variables, their distributions and expected value, and see absorption-extraction formula and van der Monde identity applied, with inverse image partitions and indicator functions shaping the distribution concept.

Explore the distribution of a random variable using a concrete dice-sum example, showing inverse images, partitioning the sample space, and computing probabilities for values two to twelve.

Describe the distribution of the random variable X, the number of spades in a five-card hand from a 52-card deck. Compute probabilities using 52 choose 5 and 13 choose k.

Understand that the expected value of a random variable is a weighted average of its values with their probabilities, illustrated by an indicator-function example where expectation equals the event probability.

Compute the expected value of a random variable by rolling a die once, where the variable equals the outcome 1–6 with probability 1/6, like the identity function.

Compute the expected value of the random variable defined by the sum of two dice, illustrating that with finite equiprobable outcomes the expectation is the arithmetic mean and equals 7.

Explore the expected value of the number of tails in six coin tosses, derive its distribution with binomial coefficients, and generalize to n tosses, yielding E[X] = n/2.

Compute the expected number of spades when drawing five cards from a 52-card deck, using van der Monde identity and absorption-extraction formulas.

Explore the absent-minded secretary problem by analyzing permutations with at least one fixed point and deriving the probability limit as n grows, approaching 1 minus 1/e.

Explore the basics of number theory within discrete mathematics, including divisors, prime factorization, greatest common divisor, and congruence modulo, with Diophantine equations and modular arithmetic.

Compute trailing zeros in 2025 factorial by tallying factors of five across powers of five, 5, 25, 125, and 625. The same 505 zeros hold for 2025-2029.

Show that for all integers n, the product n^2(n^2-1)(n^2-4) is divisible by 360 by factoring into n-2, n-1, n, n+1, n+2 and proving divisibility by 8, 9, and 5.

Prove that for n greater than 2, if one of two to the n minus one or two to the n plus one is prime, the other is composite.

apply mathematical induction to show that 7 divides 2^n + 2 + 3^{2n+1} for all n starting at 0, detailing the base case and induction step.

Explore divisibility and factoring by showing that the product of four consecutive numbers n-1, n, n+1, n+2 is divisible by 12, using 3 and 4, which are relatively prime.

Explore divisibility by 74 for all natural numbers n in problem 5 using a binomial theorem approach, showing divisibility by 2 and by 37 separately and combining the results.

Apply the fact that numbers not divisible by 3 have square congruent to 1 mod 3 to deduce p=3 when p and p^2+8 are prime, giving p^3+4=31.

Explore a playful divisibility problem (problem 7): find n and k such that n! plus 8 equals a power of two. The only solutions are (4,5) and (5,7).

Learn the sieve of Eratosthenes to find all primes up to 200 by encircling primes with p^2 ≤ n and crossing out multiples, leaving only primes.

Discover why the sieve of Eratosthenes works: start with primes up to sqrt(n), cross out multiples from p^2, and retain the uncrossed numbers as primes.

Examine and prove key divisibility properties, including transitivity, anti-symmetry, and absolute-value considerations for integers, and explore divisors of one.

Define the greatest common divisor and the least common multiple, and illustrate their use in simplifying fractions and adding fractions with a common denominator.

Compute gcd and lcm from prime factorization by using the lowest powers for common factors, the highest powers for all factors, and note gcd times lcm equals a times b.

Learn how linear combinations show that a common divisor of two integers divides all their combinations. If a combination equals one, the integers are relatively prime, foreshadowing the gcd.

Proves that positive integers a and b have a quotient q and remainder r with a = bq + r and 0 ≤ r < b, via the well-ordering principle.

Explore Euclid's algorithm for two natural numbers through divisions with remainders to find the greatest common divisor and least common multiple, applying it to diophantine equations and modular problems.

Discover how Euclid's algorithm computes the greatest common divisor via division with remainder and how the last non-vanishing remainder becomes a linear combination of the original numbers, Bezu identity.

Practice mastering Euclid's algorithm by solving eight problems, then derive the gcd as a linear combination of 29 and 24 using backward substitution, revealing coefficients m and n.

Prove that if p and q are relatively prime and both divide n, then pq divides n, using xp+yq=1 and Euclid's algorithm; sketch induction for many such divisors.

Prove lemmas about divisibility: if a prime p divides a product mk, then p divides m or k, and generalize to a product of integers using the relatively prime characterization.

Revisit the factorization theorem and prove existence and uniqueness of prime factorization for n≥2 under ordered primes, using strong induction and the divisibility lemma for primes dividing a product.

Derive the sum-of-all-divisors function sigma(n) from prime factorization, showing it equals the product over primes of (p^{alpha+1}-1)/(p-1), and illustrate a graphical, tabular proof.

Explore a really cool formula for the product of 1 minus x_i, expanding into sums of singles, pairs, triples, and the final product, linking to Viete's formulas.

Prepare for the phi function by using inclusion-exclusion to count numbers up to 60 relatively prime to 60, and preview two proofs for phi(n).

Explore Euler's totient function and count numbers up to n that are relatively prime to n. See two proofs: prime-factorization with inclusion-exclusion, and multiplicativity.

Explore linear Diophantine equations with two unknowns, learn how to use Euclid's algorithm to test solvability, find particular solutions, and generate all integer solutions.

Learn a solved problem of linear Diophantine equations using 12x+16y=2000. Apply the Euclidean algorithm to find a particular and general solution, then enforce x,y≥1 to obtain 41 valid purchases.

Analyze a Diophantine equation with a negative coefficient and x restriction between 0 and 21. Use Euclid’s algorithm to find a starting solution, yielding only two solutions: (9,4) and (20,9).

Explore how straight lines in the plane are described by slope-intercept and standard forms, derive rise over run, and convert to Diophantine equations to find integer points.

Explore diophantine equations through geometric interpretation, solving three diophantine equations to find all integer points on lines, using gcd, particular solutions, and parametric general solutions.

Master linear Diophantine equations through Euclid's algorithm, deriving particular and general solutions using coefficients 24 and 29, with x = -12 + 29k and y = 10 - 24k.

Explain the decimal system as a base-10 positional system with digits 0–9. Explore how each position expresses units, tens, hundreds, and the role of zero.

Explore the binary positional system (base 2), using digits 0 and 1, learn binary–decimal conversions, division-by-two method, and why remainders yield the correct binary representation.

explains positional number systems for bases 2, 4, 8, and 16 (hexadecimal), and demonstrates converting between bases using grouping and remainders, with decimal and binary context.

Compare numbers across bases by digit count and left-to-right digits to determine order. Convert to a common base, especially binary, to practice conversions between base 8, base 4, and decimal.

Explore arithmetic in binary and other bases in discrete mathematics 2, mastering addition, subtraction, and multiplication while comparing decimal conversion with native base methods and validating results.

Practice arithmetic in various positional number systems by converting quaternary numbers to binary, multiplying in binary, and converting the result to base 4, 8, and 16.

Delve into rational and irrational numbers, their decimal expansions, and density in real numbers. Learn Cantor's diagonal argument showing the interval between 0 and 1 is uncountable.

Explore why 0.999... equals 1 in decimal and base-b representations, with infinite periodic expansions, geometric series, and the difference of nth powers.

Learn modular arithmetic by counting and computing modulo n, using remainders and division with remainder, and solve linear congruences while exploring equivalence classes and invertibility concepts.

Explore modular arithmetic foundations, including the equivalence relation of congruence modulo n, residue classes, complete residue systems, and Zn, through division with remainder examples.

Explore the basic properties of modular arithmetic, including translation, scaling, addition, subtraction, multiplication, exponentiation, and polynomial congruences, with proofs and practical examples.

Explore addition and multiplication in Zn, using representatives of classes, and understand their properties, inverses, and visual illustrations in Zn, z5, and z6.

Derives and explains divisibility tests for 2, 4, 8, 16, 5, and 10 using modular arithmetic in the decimal system, showing how to apply them with last digits.

Demonstrate divisibility tests by 3, 9, 6, 12, and 15 through digit-sum criteria and two proofs, including a polynomial approach linking x to its digits.

Show how to test divisibility by 12 using tests for 3 and 4, via sum of digits and last two digits, noting they are relatively prime, with a watch-time example.

Explore the divisibility by 9 and 3 using the sum of digits and modular arithmetic, and apply sidewise multiplication to show a product is incorrect, via casting out nines.

Learn a divisibility test for 11 using an alternating sum of digits from the last digit. It reveals divisibility and the remainder modulo 11.

Derive divisibility tests by 7, 13, and 14 using three-digit groupings and alternating sums; show that 1001 equals 7×11×13 and the 14 test combines 7 with 2 to reveal remainders.

In discrete mathematics 2, learn to solve congruences by using ones and minus ones to find remainders of large powers, illustrated with 49^3857 mod 25.

Learn to solve congruences efficiently by exploiting powers modulo 7: find that 3^3 ≡ -1 mod 7, raise to the power, and obtain 3^31 ≡ 3 mod 7.

Explore solving congruences by seeking ones or minus ones modulo 7, replacing 17 with 3 to simplify, use 3^3 ≡ -1 and transitivity to get 17^64 ≡ 4 (mod 7).

Reduce 47^109 modulo 15 by replacing 47 with 2, show 2^4 ≡ 1 (mod 15), then 2^109 ≡ 2 (mod 15), so the remainder is 2.

Learn to solve congruences by finding powers equal to 1 or minus 1 modulo 5, illustrated with 32^19 mod 5. Compare two methods to arrive at 3.

Revisiting three classic divisibility problems, this lecture demonstrates using congruences and modular arithmetic to prove divisibility by 7, 11, and 133 via sums, differences, and common factors without induction.

Explore revisiting three divisibility problems using congruences, comparing induction and binomial theorem, and uncover repeating patterns in powers to prove divisibility by 74, 9, and related numbers.

Examine two cancellation properties for congruences, including canceling added or subtracted terms, and canceling a common factor only when gcd(n, c)=1. Introduce a supporting lemma that justifies the second property.

Explore Fermat's little theorem: formulate a^p ≡ a modulo p for prime p, its corollary for integers relatively prime to p, and cancellation in congruences with practical examples.

Explore Fermat's little theorem with a proof by induction and the binomial theorem, showing that a^p ≡ a (mod p) for prime p and all integers a.

This combinatorial proof of Fermat's little theorem shows that for prime p and positive a, a^p minus a is divisible by p by grouping non-monochrome necklaces into p rotational classes.

Explore the third proof of Fermat's little theorem and Euler's totient theorem through a lemma showing a times S is a permutation of S modulo p, yielding a^(p-1) ≡ 1.

Apply Fermat's little theorem and congruences to discover patterns in n^k modulo m (2–9, 1–5) and demonstrate divisibility results, including n^7−m divisible by 42, using factoring and cyclic patterns.

Explore solving linear equations in real numbers and in Zn, using cancellation and inverses, with concrete examples in Z6 and prime moduli.

Solve linear equations in Zn using substitution and graphical representations, demonstrated with 4x ≡ 1 mod 5; x = 4 is the unique solution validated by enumeration and visualization.

Solve the linear equation 3x plus 5 equals 7 in Z8 by checking all elements of Z8, simplify to 3x equals 2, and verify x equals 6.

Factor 72 as 2^3·3^2 and 120 as 2^3·3·5; conclude x must include the missing factor 5, so x ∈ {0,5,10,…,115} with 24 solutions in Z_120.

Learn how the zero product property fails in Z_n for composite n, with counterexamples in Z6, Z8, and Z15, and why it holds in Z_p for prime p.

Explore invertible elements in Zn, show gcd(a,n)=1 criterion, compute inverses with Euclid's algorithm, solve linear congruences, and note Zp all nonzero elements invertible.

Identify invertible elements in Z9 and Z20 by being relatively prime to n, build a Z9 multiplication table, and illustrate Euler's totient theorem using set S and inverses.

Discover how to find a multiplicative inverse in zn using Euclid's algorithm, verify invertibility via gcd, and compute 89's inverse in z210 through a linear combination.

Explore invertible elements in Zn through a Diophantine problem, showing how 10x+3=5 in Z22 reduces to 10x≡2 (mod 22) with solutions x=9 or 20.

Reformulate diophantine problems into Zn equations, apply Euclid's algorithm and congruences, and practice solving restricted x and y by translating prior problems into modular forms.

Explore solving systems of linear equations in real numbers and Zn using elimination and opposite coefficients, with attention to cancellations and differences between R and Zn.

solve systems of linear equations in Z7 using elimination and inverse methods, demonstrating two approaches and verifying the solution x=1, y=4 modulo 7.

solve a system of linear equations in z25 using opposite coefficients to eliminate y, yielding a unique solution x=5, y=20 and verifying the result in modulo 25.

Learn how to solve a system of congruences using the Chinese remainder theorem, illustrated with moduli 5, 6, and 7, and inverses in Z5, Z6, Z7, with cryptography context.

Solve a set of simple quadratic equations in Zn, including x^2 ≡ 0 (mod 18) with solutions 0, 6, and 12, and contrast solving in Zn with solving in R.

Explore solving a quadratic equation in Z11 by exhaustively testing elements 0 through 10, revealing solutions x=3 and x=5 and introducing modular arithmetic problems.

Explore creative divisibility challenges by counting natural numbers under 10,000 divisible by 6 using digits 0, 1, and 2; apply last-digit and digit-sum rules to find 18 solutions.

Explore fun divisibility problems by examining the last digits of n! sums S_n, showing only n=1 and n=3 yield a perfect square.

Apply the divisibility by four test to numbers formed by ones; they are congruent to three modulo four, while odd squares are one modulo four, so none are squares.

Use congruences modulo 10 to compare last digits and test divisibility by 10. See how exponent modulo 4 determines last-digit cycles in powers, enabling a difference check for 10.

Show that a power of two with 1000 digits cannot have each digit 0-9 exactly 100 times, using divisibility by 3 and the sum of digits to reach a contradiction.

Show that the sum p^2+q^2+r^2 of three consecutive primes is prime only for p=3, q=5, r=7, with divisibility by 3 explaining why.

Apply the cycle of remainders for powers of 10 modulo 7, of length 6; since 100 ≡ 4 modulo 6, conclude 10^100 ≡ 4 modulo 7.

Explore a fun divisibility problem by converting 837 to base 3 and using ternary representation. Determine that K, M, and N are 6, 4, and 3 in order, via permutations.

Discover the repeating pattern of the last two digits of powers of seven every four terms and use it to sum a1 through a2026, yielding 50656.

Explore discrete mathematics divisibility by finding all n ≤ 400 with nine divisors, using the prime factorization formula to show n is either p^8 or p^2 q^2.

Analyze 25^n + 9^n divisible by 13 using congruences; reveal a 3-cycle for 9^n and show divisibility occurs for odd n multiples of 3, yielding 338 solutions below 2026.

Prove that n^(n-1) - 1 is divisible by (n-1)^2 when n > 1 using binomial theorem with m = n-1, with small-n checks and a proof for n >= 5.

Solve a challenging divisibility problem by 11 using the alternating sum test, Fermat's little theorem, and power rules to show the remainder is 1.

Explore the entrance to abstract algebra, focusing on fundamental structures like groups, rings, fields, and vector spaces, and see how number theory informs algebraic questions.

Explore how mathematicians build abstract models from concrete examples, and examine algebraic structures such as groups, rings, fields, vector spaces, metrics, and isometry groups.

Explore the group as a non-empty set with a binary operation, covering its order, subgroups, and axioms—closure, associativity, and inverse—plus multiplicative vs additive notation and abelian versus non-abelian.

examine the first natural examples of groups by testing number sets under addition and multiplication. identify closure, inverses, and neutral elements, with counterexamples and subgroup relationships.

Explore finite groups from addition and multiplication modulo n: Zn has additive inverses, while Z_p without 0 forms a multiplicative group when p is prime.

Explore core group properties, including uniqueness of the neutral element, uniqueness of inverses, inverse of the inverse, and the sock shoe property for products, with illustrations.

Study three operations lacking nice properties: diamond, square, and star. Diamond is commutative but not associative; square and star fail commutativity and associativity.

Explore Alon Amit's neat trick that rewrites the circle operation x circle y as addition after a 45 shift and reciprocal, proving its commutativity and associativity.

Investigate a strange operation, circle: x circle y = x + y − xy; prove associativity and commutativity, identify 0 as neutral, derive inverse x/(x−1) for x ≠ 1.

Explore Klein's four group, a small but exciting abelian group with four elements where each element is its own inverse, and learn to construct its Cayley table.

Explore line symmetries and rotations as plane isometries that preserve distances, illustrate reflection across a symmetry line, and rotate points around a center with fixed points.

Examine groups of permutations, the symmetric groups S_n, and how composition of bijections yields non-commutative groups. Explore triangle isometries, Cayley tables, and the idea of isomorphism linking these structures.

Explore the order of elements in groups, analyze cyclic groups, and examine their subgroups to understand foundational structure in abstract algebra.

Explore cyclic groups through additive Zn and multiplicative Cn, highlighting their isomorphism via rotations. See how composition mirrors addition modulo n and generators in Z3, Z5, and Z7 without 0.

Identify the group of units U9 as the invertible elements of Z9 under multiplication modulo 9 and prove its group properties, including closure, inverses, and cyclicity generated by 2.

Show the group of units U18 inside Z18 under multiplication mod 18, identify its elements, and note the cyclic subgroup generated by 7 has order 3 and inverses.

Explore fields as sets with addition and multiplication, forming additive and multiplicative abelian groups, with distributivity, where R and Q are fields and Z_p is a field for prime p.

Show that for any non-square D, the field {alpha + beta sqrt(D) : alpha, beta in Q} lies between Q and R, with sqrt(D) irrational and D yielding distinct fields.

This lecture derives complex numbers as pairs of real numbers with addition and a multiplication, showing i = (0,1) and i^2 = -1, and that the field cannot be ordered.

Explore the definition of rings, their axioms, and familiar examples such as Z, R, Q, and Zn, plus a non-commutative matrix ring, highlighting commutativity and distributivity.

Define the direct product of two groups as their Cartesian product with coordinate-wise operations, preserving associativity and inverses, and show rings formed this way may fail to be fields.

Explains homomorphisms and isomorphisms between structures, showing how bijections that preserve operations create equivalent algebraic structures across fields, rings, and groups.

Explore how Z6 and Z7 without 0 form isomorphic groups of order 6 by constructing a one-to-one homomorphism, sending 0 to 1 and 1 to the generator 3.

Explore the preparations for the Chinese remainder theorem, as referenced by V174, in the context of discrete mathematics.

Examine how vector spaces generalize vector addition and scalar multiplication through diverse examples—matrices, functions, real and complex numbers—highlighting linear combinations and axioms.

Explore how Klein's four-group, the direct product Z2×Z2, and the group of own isometries of a non-square rectangle share the same structure, via Cayley tables and explicit isomorphisms.

Tackle an optional, difficult problem about isomorphic groups, guiding students to apply group structure and isomorphism reasoning to determine mappings and equivalences.

Compare four-element Cayley tables to determine which define a group, identify two groups and two non-groups, and explain why the two groups are not isomorphic.

Explore the dihedral group D3, the isometries of an equilateral triangle, and its isomorphism with the symmetric group S3 through rotations and reflections.

Explore the dihedral group D4, the eight isometries of a square, including the identity, four line symmetries, and three rotations, and see its non-abelian structure as a subgroup of S4.

Explore subgroups, cosets, and Lagrange's theorem to understand structure and order in finite groups using discrete mathematics.

Explore dihedral groups D3 and D4 and their relation to symmetric groups S3 and S4 via the pictures. Examine rotations, odd and even permutations, cosets, Lagrange's theorem, and four cycles.

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