Summary of "[문제 풀이] 중1 수학 (상) 1단원. 소인수분해 - [진격의홍쌤]"

Overview

Topic: Prime factorization and related problems for 1st-year middle-school math (basic number theory).

Goals: learn how to factor numbers into primes, represent prime factorization with exponents, count divisors, find greatest common divisors (GCD), test for coprimality, and find least common multiples (LCM) — including for algebraic expressions.

Teaching approach: teacher works through textbook/workbook problems, shows multiple methods for factorization, emphasizes understanding the principles and practicing problems before watching the lecture.

Key concepts

Prime number: an integer greater than 1 whose only positive divisors are 1 and itself.
Composite number: an integer greater than 1 that has divisors other than 1 and itself.
Special case: 1 is neither prime nor composite.
Prime factorization: express a number as a product of primes. Example: 60 = 2^2 · 3 · 5.
Number of positive divisors d(n): if n = p1^a · p2^b · … then d(n) = (a + 1)(b + 1)….
- Example: if n = p^3 · q^2, then d(n) = (3 + 1)(2 + 1) = 4 × 3 = 12.
Greatest common divisor (GCD): for common prime bases choose the smaller exponent and multiply those primes.
Least common multiple (LCM): for each prime/variable choose the larger exponent among the expressions and multiply them together.
Coprime (relatively prime): two integers whose GCD = 1 (they share no prime factors). If both are multiples of the same prime (e.g., both multiples of 3), they are not coprime.

Methods — how to find prime factorization (example: 60)

Successive splitting
- Split into two factors repeatedly:
  - 60 = 2 × 30, 30 = 2 × 15, 15 = 3 × 5.
  - Collect primes: 60 = 2 × 2 × 3 × 5 = 2^2 · 3 · 5.
Factor tree / pruning method
- Start with 60, factor into two factors (e.g., 6 × 10), factor each branch until all leaves are prime.
- Combine leaves to get 2^2 · 3 · 5.
Repeated division by primes
- Divide 60 by the smallest prime (2) repeatedly until remainder 0, then continue with the next primes until the quotient is 1.
- Result: 2^2 · 3 · 5.

Tip: represent factors using exponents for compactness (e.g., 2^2 · 3 · 5).

Counting divisors

Formula: if n = p1^a · p2^b · … then the number of positive divisors is d(n) = (a + 1)(b + 1)….
Example: n = p^3 · q^2 ⇒ d(n) = (3 + 1)(2 + 1) = 12.

Finding the GCD

Steps: 1. Prime-factor each number. 2. For each prime that appears in both factorizations, take the smaller exponent. 3. Multiply those primes raised to the chosen exponents. - Ignore primes that are not shared.

Deciding coprimality

Two numbers are coprime if their GCD = 1 (i.e., they share no prime factors).
Quick check: if both numbers are multiples of the same prime, they cannot be coprime.

Finding the LCM (numbers or algebraic expressions)

Steps: 1. Prime-factor each number or expression (treat variables as bases). 2. For each base (prime or variable), take the maximum exponent appearing in any factorization. 3. Multiply those together to get the LCM. - For algebraic LCM, compare exponents for each variable separately (e.g., take a^max, b^max). Numeric coefficients’ prime exponents are handled the same way. - Teacher tip: write variables in a consistent order and organize terms by base to avoid mistakes.

Problems covered (examples solved in class)

Multiple-choice: identify the incorrect prime-factorization among choices (answer: choice 2).
Distinguish prime vs composite numbers; reminder that 1 is neither.
Prime-factorize 60 using three different methods (example).
Compute the number of divisors from prime-exponent form using the (exponent + 1) product rule.
Find GCD by taking common primes with smaller exponents.
Choose which pairs are coprime by checking shared prime factors.
Find LCM of algebraic expressions by taking highest exponents for each base/variable (teacher computed an example that led to a + b = 7).

Teaching tips and recommendations

Try solving problems yourself before watching the solution.
When working with algebraic expressions, write variables in a consistent, organized way.
Use whichever factorization method you find most convenient.
Focus on understanding the underlying principles rather than memorizing procedures.
If something is unclear, rework problems until you understand the principle and ask questions (e.g., in comments).

Speakers / sources

Main speaker / instructor: the teacher (channel/title: 진격의홍쌤).
References: textbook/workbook problems used in class (no specific external sources named).
Intended audience: first-year middle-school students.