Summary of "สรุป "ทฤษฎีจำนวน" (ตอนที่ 2) | เตรียมความพร้อมสู่การสอบ สอวน. คอมพิวเตอร์ ค่าย 1"

Main Ideas, Concepts, and Lessons Conveyed

Number Theory is frequently tested in math competitions and exams like สอวน. (Thailand's Olympiad in Informatics).
Although some topics are not in the standard curriculum, they appear often in competitions.
The tutor encourages students to practice and understand division, remainders, prime numbers, and related concepts.

Fundamental division formula: Dividend = Divisor × Quotient + Remainder
Problems involving prime divisors where two numbers give the same remainder upon division.
Using variables for quotient and remainder to form equations.
Subtracting equations to find relationships and divisibility (e.g., divisor divides the difference of dividends).

Definitions and properties of GCD and LCM.
GCD divides both numbers evenly.
LCM is related to GCD by the formula: LCM(a, b) × GCD(a, b) = a × b
Euclidean Algorithm for finding GCD:
- Repeated division with remainder.
- Last non-zero remainder is the GCD.
Linear combinations: expressing GCD as a linear combination of two numbers (ax + by = GCD).
Applications in solving Diophantine Equations.

Use of the Remainder Theorem: The remainder when a polynomial f(x) is divided by (x - a) is f(a).
Substituting values into polynomials to find remainders.
Examples with cubic polynomials and given coefficients.

Two numbers are relatively prime if their GCD is 1.
Properties of relative primes:
- Multiplying relative primes keeps the GCD as 1.
- Powers of relative primes remain relative prime.
Problems involving counting numbers relatively prime to a given number within a range.
Use of inclusion-exclusion principle to count multiples and non-multiples.

Definition of congruence: a ≡ b (mod m) means a and b leave the same remainder when divided by m.
Properties of Modular Arithmetic:
- Addition and multiplication of congruences.
- Powers under modulo.
Using Modular Arithmetic to solve problems such as finding units digits and remainders of large exponents.
Examples:
- Units digit of 8^50.
- Remainder of 3^2556 when divided by 10.
Application of Modular Arithmetic in simplifying large exponent calculations.

Step-by-step solving of past exam problems from various years (2013, 2014, 2016, 2017, 2020, etc.).
Emphasis on understanding the properties and applying the Euclidean Algorithm.
Encouragement to practice linear combinations and Modular Arithmetic.
Advice on recognizing patterns and using theorems to simplify calculations.
Importance of accurately finding remainders and divisors in problems.

Announcement of future tutoring sessions (both online and onsite).
Mention of supplementary videos for deeper understanding, especially on Modular Arithmetic.
Encouragement to visit a specific website (SJC Computer) for more practice and video tutorials.
Teacher Job’s teaching style includes humor, interaction, and encouragement.

Using the Division Algorithm:
- Express dividend as divisor × quotient + remainder.
- Set variables for unknown quotients and remainders.
- Subtract equations to find divisibility conditions.
Finding GCD via Euclidean Algorithm:
- Divide larger number by smaller number.
- Replace larger number with smaller number and smaller number with remainder.
- Repeat until remainder is zero; last non-zero remainder is the GCD.