Discuss Types of Numbers and Factors of a Number. Include explanations, examples, differentiation, methods to find factors, and identification of prime/composite numbers.

Types of Numbers

Explanation of 4 Types of Numbers:

1. Even numbers: Integers divisible by $2$ with remainder $0$.
   Examples: $2, 4, 6, 8, 10$.

2. Odd numbers: Integers not divisible by $2$.
   Examples: $1, 3, 5, 7, 9$.

3. Prime numbers: Natural numbers greater than $1$ with exactly $2$ distinct positive factors ($1$ and itself).
   Examples: $2, 3, 5, 7, 11$.

4. Composite numbers: Natural numbers greater than $1$ that are not prime (more than $2$ positive factors).
   Examples: $4, 6, 8, 9, 10$.

Examples and Differentiation:

• $2$: Even and prime.
• $3$: Odd and prime.
• $4$: Even and composite (factors: $1,2,4$).
• $9$: Odd and composite (factors: $1,3,9$).

Numbers can have multiple types (e.g., $6$ is even and composite).

Factors of a Number

Definition and Explanation of Factors:
Factors of a number $n$ are positive integers $d$ such that $n \div d$ gives quotient $q$ with remainder $0$ ($n = d \times q$).
Example for $n=12$:
Factors: $1,2,3,4,6,12$.

Method to Find Factors:
Use pair method up to $\sqrt{n}$. Example: $n=36$, $\sqrt{36}=6$.

Step 1: List divisors from $1$ to $\sqrt{n}$. Formula: Check if $d$ divides $36$ ($36 \mod d = 0$), add pair $36/d$.
Step 2: $d=1$: $36 \div 1 = 36$ (pair: $1,36$).
Step 3: $d=2$: $36 \div 2 = 18$ (pair: $2,18$).
Step 4: $d=3$: $36 \div 3 = 12$ (pair: $3,12$).
Step 5: $d=4$: $36 \div 4 = 9$ (pair: $4,9$).
Step 6: $d=5$: $36 \mod 5 = 1 \neq 0$ (skip).
Step 7: $d=6$: $36 \div 6 = 6$ (pair: $6$).
Factors: $1,2,3,4,6,9,12,18,36$.

Differences between Factors and Multiples:

• Factors of $n$ divide $n$ ($d \mid n$, remainder $0$). Example for $6$: $1,2,3,6$.
• Multiples of $n$ are $n \times k$ ($k$ positive integer). Example for $6$: $6,12,18,24$.
  Key: $m$ is multiple of $n$ iff $n$ is factor of $m$.

Identification of Prime and Composite Numbers:
Prime: $>1$, exactly $2$ distinct positive factors ($1$ and itself).
Composite: $>1$, $>2$ positive factors.
$1$: $1$ factor (neither).

Examples: