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

Brief overview

This lesson covers the first middle‑school math unit: prime numbers, composite numbers, prime factorization, greatest common divisor (GCD), coprime numbers, and least common multiple (LCM). It explains definitions, gives examples, shows methods for computing prime factorizations, GCD, and LCM, and ends with study advice.

Key definitions and concepts

Natural numbers: positive integers 1, 2, 3, …
Prime number: a natural number whose only divisors are 1 and itself (examples: 2, 3, 5, 7).
Composite number: a natural number with divisors other than 1 and itself (examples: 4, 6, 8, 9, 10).
Factor (divisor): a number that divides another number exactly (e.g., factors of 6: 1, 2, 3, 6).
Prime factor: a factor that is prime (prime factors of 6: 2 and 3).
Prime factorization: expressing a number as a product of prime numbers (e.g., 36 = 2 × 2 × 3 × 3 = 2^2 × 3^2).
Common divisor: a divisor shared by two (or more) numbers.
Greatest common divisor (GCD): the largest common divisor of given numbers.
Coprime (relatively prime): two numbers whose GCD is 1 (example: 2 and 3).
Common multiple: a multiple shared by two (or more) numbers.
Least common multiple (LCM): the smallest positive common multiple (example: LCM(4, 6) = 12).

Note: The number 1 is special — it is neither prime nor composite.

Methods

1) Prime factorization — three equivalent ways (example: 36)

Choose whichever method feels fastest and clearest to you.

Method A — successive factoring (step-by-step)
1. 36 = 2 × 18
2. 18 = 2 × 9 → 36 = 2 × 2 × 9
3. 9 = 3 × 3 → 36 = 2 × 2 × 3 × 3
4. With exponents: 36 = 2^2 × 3^2
Method B — factor tree / pruning visualization
- Split 36 into factors repeatedly (e.g., 36 → 2 × 18, 18 → 2 × 9, 9 → 3 × 3).
- Collect the prime leaves: 2, 2, 3, 3 → 36 = 2^2 × 3^2
Method C — repeated division (division ladder)
- Divide by prime factors in sequence:
  - 36 ÷ 2 = 18
  - 18 ÷ 2 = 9
  - 9 ÷ 3 = 3
  - 3 ÷ 3 = 1
- Primes used: 2, 2, 3, 3 → 36 = 2^2 × 3^2

2) Greatest common divisor (GCD) — two methods (example: 12 and 18)

Method A — listing factors (good for small numbers)
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common divisors: 1, 2, 3, 6 → GCD = 6
Method B — prime factorization / take minimum exponents
- 12 = 2^2 × 3^1
- 18 = 2^1 × 3^2
- For each prime present in both, take the smaller exponent:
  - 2: min(2,1) = 1 → 2^1
  - 3: min(1,2) = 1 → 3^1
- GCD = 2^1 × 3^1 = 6
Division-based algorithms (Euclidean algorithm)
- Faster for large numbers; repeatedly use division/remainder to find the GCD.
Coprime check
- If GCD = 1, the numbers are coprime (e.g., 2 and 3).

3) Least common multiple (LCM) — two methods (example: 4 and 6)

Method A — list multiples (good for small numbers)
- Multiples of 4: 4, 8, 12, 16, 20, …
- Multiples of 6: 6, 12, 18, 24, …
- Smallest common multiple = 12 → LCM = 12
Method B — prime factorization / take maximum exponents
- 4 = 2^2
- 6 = 2^1 × 3^1
- For each prime present in either number, take the larger exponent:
  - 2: max(2,1) = 2 → 2^2
  - 3: max(0,1) = 1 → 3^1
- LCM = 2^2 × 3^1 = 12
Division-ladder method
- Divide the pair of numbers by common primes stepwise. Multiply the primes used and any remaining numbers to get the LCM — this is equivalent to collecting highest prime powers.

Tips, relationships, and study advice

Remember the exception: 1 is neither prime nor composite.
Choose the method that fits the problem size:
- Listing factors/multiples is fine for small numbers.
- Prime factorization (min/max exponents) is systematic and scales to larger numbers.
- Euclidean-type division is efficient for large numbers when computing GCD.
Practice is essential: solve many problems to internalize methods and discover shortcuts.
The teacher noted the EBS textbook covers this topic across multiple lectures — it’s a sizable unit, so study slowly and review often.

Speakers / sources

Instructor/Narrator: the teacher presenting the lesson (진격의홍쌤 — “Hong teacher”)
Referenced source: EBS textbook/lecture (mentioned by the teacher)
Background: non‑spoken background music noted at the end of the subtitles

Next steps

If you’d like, I can: - Produce a one‑page quick reference sheet (definitions + step‑by‑step recipes) you can print, or - Create a few worked practice problems (with solutions) using these methods.

Which would help most?