1. **Problem Statement:** Rearrange and solve the given multi-part math questions involving rational numbers, set operations, divisibility, properties of numbers, and algebraic simplifications. --- ### PART I: True or False Statements 1. Every rational number can be written as a fraction of two integers. - This is **True** by definition of rational numbers. 2. There is no even prime number. - This is **False** because 2 is an even prime number. 3. The sum of any two irrational numbers is an irrational number. - This is **False**; for example, $\sqrt{2}$ and $-\sqrt{2}$ are irrational but their sum is 0 (rational). 4. If the numerator is less than the denominator, then it is called a proper fraction. - This is **True** by definition. 5. Dividing two numbers with the same sign always gives a positive result. - This is **True** because positive divided by positive or negative divided by negative is positive. --- ### PART II: Set Operations Matching Given: $$U=\{1,2,3,4,5,6,7,8,9,10\}, A=\{2,3,5,7\}, B=\{2,3,4,6,8,10\}$$ Recall: - $A \Delta B = (A \setminus B) \cup (B \setminus A)$ (symmetric difference) - $B^\prime = U \setminus B$ - $A^\prime = U \setminus A$ Calculate: 1. $A \Delta B = \{5,7,4,6,8,10\}$ (elements in A or B but not both) 2. $A \cap B^\prime = A \cap (U \setminus B) = \{5,7\}$ 3. $B^\prime = U \setminus B = \{1,5,7,9\}$ 4. $A^\prime \cap B^\prime = (U \setminus A) \cap (U \setminus B) = U \setminus (A \cup B) = \{1,9\}$ 5. $U \setminus A \Delta B = U \setminus \{5,7,4,6,8,10\} = \{1,2,3,9\}$ Match with Column B: - 1. $A \Delta B$ matches E: $\{4,5,6,7,8,10\}$ (note slight difference, but closest) - 2. $A \cap B^\prime$ matches A: $\{5,7\}$ - 3. $B^\prime$ matches D: $\{1,5,7,9\}$ - 4. $A^\prime \cap B^\prime$ matches B: $\{1,9\}$ - 5. $U \setminus A \Delta B$ matches C: $\{1,2,3,9\}$ --- ### PART III: Multiple Choice Questions 1. Which statement is correct? - 1 is either odd or composite: **True** (1 is neither prime nor composite, so false) - 4 is the only even prime number: **False** (4 is not prime) - 343 divisible by 3: **False** (sum digits 3+4+3=10 not divisible by 3) - 91 is composite: **True** (91=7×13) 2. If $36=3 \times 12$, which is correct? - 36 is a factor of 12: False - 12 is divisible by 36: False - 3 is divisible by 36: False - 36 is divisible by 3: **True** 3. Which is correct? - $-\frac{1}{2} \in W$ (Whole numbers): False - $\frac{4}{5} \in Z$ (Integers): False - $2.65 \in Q$ (Rational numbers): **True** - $4 \in Z^-$ (Negative integers): False 4. Divisible by 9? - 572: sum digits 5+7+2=14 not divisible by 9 - 49: sum 4+9=13 no - 288: sum 2+8+8=18 divisible by 9 **True** - 188: sum 1+8+8=17 no 5. Correct statement? - Addition closed under irrational numbers: False (sum can be rational) - Division closed under rational numbers: False (division by zero undefined) - Subtraction closed under irrational numbers: False (difference can be rational) - Multiplication closed under rational numbers: **True** --- ### PART IV: Fill in the blanks 1. For any two real numbers $a$ and $b$, one of $a < b$, $a = b$, or $a > b$ holds. This is called the **Trichotomy Law**. 2. A natural number with exactly two distinct factors (1 and itself) is called a **Prime Number**. 3. A decimal number that is neither terminating nor repeating is called an **Irrational Number**. 4. A composite number expressed as a product of prime numbers is called the **Prime Factorization**. 5. If 12 is divided by 5, the remainder is **2**. --- ### PART V: Work Out 1. Find LCM and GCF of 8, 12, 18, 24. - Prime factorizations: - $8 = 2^3$ - $12 = 2^2 \times 3$ - $18 = 2 \times 3^2$ - $24 = 2^3 \times 3$ - GCF: take minimum powers of common primes: - $2^{\min(3,2,1,3)} = 2^1 = 2$ - $3^{\min(0,1,2,1)} = 3^0 = 1$ - So, GCF = $2 \times 1 = 2$ - LCM: take maximum powers: - $2^{\max(3,2,1,3)} = 2^3 = 8$ - $3^{\max(0,1,2,1)} = 3^2 = 9$ - So, LCM = $8 \times 9 = 72$ 2. Convert repeating decimal $4.2\overline{02}$ to fraction. Let $x = 4.202020...$ Multiply by 100 (two repeating digits): $$100x = 420.202020...$$ Subtract original: $$100x - x = 420.202020... - 4.202020... = 416$$ So, $$99x = 416 \Rightarrow x = \frac{416}{99}$$ 3. Simplify: A. $8\sqrt{24} + \frac{2}{3} \sqrt{54} - 2\sqrt{96}$ - Simplify radicals: - $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ - $\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$ - $\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$ Substitute: $$8 \times 2\sqrt{6} + \frac{2}{3} \times 3\sqrt{6} - 2 \times 4\sqrt{6} = 16\sqrt{6} + 2\sqrt{6} - 8\sqrt{6}$$ Combine: $$ (16 + 2 - 8) \sqrt{6} = 10 \sqrt{6}$$ B. $(\sqrt{5} - 2)^2 + 4\sqrt{5}$ Expand square: $$(\sqrt{5})^2 - 2 \times 2 \times \sqrt{5} + 2^2 + 4\sqrt{5} = 5 - 4\sqrt{5} + 4 + 4\sqrt{5}$$ Combine like terms: $$ (5 + 4) + (-4\sqrt{5} + 4\sqrt{5}) = 9 + 0 = 9$$ 4. In a class of 31 students, 22 study Math, 20 study Physics, and 5 study neither. - Total students studying Math or Physics or both: $$31 - 5 = 26$$ - Use formula for union: $$|M \cup P| = |M| + |P| - |M \cap P|$$ Substitute: $$26 = 22 + 20 - |M \cap P|$$ Solve: $$|M \cap P| = 22 + 20 - 26 = 16$$ **Answer:** 16 students study both subjects. --- **Final answers:** - PART I: True, False, False, True, True - PART II: 1-E, 2-A, 3-D, 4-B, 5-C - PART III: 91 composite, 36 divisible by 3, 2.65 rational, 288 divisible by 9, multiplication closed under rationals - PART IV: Trichotomy Law, Prime Number, Irrational Number, Prime Factorization, 2 - PART V: GCF=2, LCM=72; fraction $\frac{416}{99}$; simplified expressions $10\sqrt{6}$ and 9; 16 students both subjects