This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 ÷ d gives quotient q with remainder 0 (n = d × q). Example for n=12: Factors: 1,2,3,4,6,12. 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1. 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 d = 0), add pair 36/d. Step 2: d=1: 36 ÷ 1 = 36 (pair: 1,36). Step 3: d=2: 36 ÷ 2 = 18 (pair: 2,18). Step 4: d=3: 36 ÷ 3 = 12 (pair: 3,12). Step 5: d=4: 36 ÷ 4 = 9 (pair: 4,9). Step 6: d=5: 36 5 = 1 ≠ 0 (skip). Step 7: d=6: 36 ÷ 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 n, remainder 0). Example for 6: 1,2,3,6. Multiples of n are n × 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: &Number 2: factors 1,2 && (prime)\\ &Number 4: factors 1,2,4 && (composite)\\ &Number 7: factors 1,7 && (prime)\\ &Number 15: factors 1,3,5,15 && (composite)