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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Step 1: Analyze Question 9. We need to evaluate the validity of each statement as a divisibility rule. Statement I: An integer k is divisible by 4 if and only if the sum of the digits of k is divisible by 4. This statement is incorrect. For example, consider the number k=12. The sum of its digits is 1+2=3, which is not divisible by 4. However, 12 is divisible by 4. The correct rule for divisibility by 4 is that the number formed by its last two digits must be divisible by 4. Statement II: An integer k is divisible by 10 if and only if the number formed by the last 3 digits of k is divisible by 10. This statement is correct. If an integer k is divisible by 10, its last digit must be 0. If the last digit is 0, then the number formed by its last 3 digits (e.g., d_2 d_1 0) will also end in 0, and thus be divisible by 10. Conversely, if the number formed by the last 3 digits of k is divisible by 10, it implies that the last digit of k must be 0. If the last digit of k is 0, then k itself is divisible by 10. Statement III: An integer k is divisible by 3 if and only if the number formed by the last 2 digits is divisible by 3. This statement is incorrect. For example, consider the number k=123. The number formed by its last two digits is 23, which is not divisible by 3. However, 123 is divisible by 3 (since the sum of its digits 1+2+3=6 is divisible by 3). The correct rule for divisibility by 3 is that the sum of its digits must be divisible by 3. Only statement II is a valid way to verify divisibility by 10. Therefore, none of the options A, B, C, or D are correct, as they all include I or III. The correct option is E. E Step 2: Analyze Question 10. We are given a=24 and b=36. We need to find the greatest common divisor (GCD) of a and b. First, find the prime factorization of each number: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2^3 × 3^1 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2^2 × 3^2 To find the GCD, we take the lowest power of each common prime factor: (24, 36) = 2^(3,2) × 3^(1,2) (24, 36) = 2^2 × 3^1 (24, 36) = 4 × 3 (24, 36) = 12 The greatest common divisor of 24 and 36 is 12. The correct option is C. C Step 3: Analyze Question 11. The question asks about the base case typically used when proving a statement by mathematical induction. Mathematical induction is a method used to prove that a statement P(n) is true for all natural numbers n (or for all n greater than or equal to some initial integer). It consists of two steps: 1. Base Case: Prove that the statement P(n) is true for the initial value of n (e.g., n=1 or n=0). 2. Inductive Step: Assume that P(k) is true for an arbitrary natural number k n_0 (the inductive hypothesis), and then prove that P(k+1) is also true. Let's evaluate the given options: A. The statement is assumed to be true for all n N. This is incorrect; we aim to prove* it for all n, not assume it. B. The statement is true for the smallest value of n. This describes the base case, but option C is more complete. C. The statement is proven true for the initial value, usually n=1 or n=0. This accurately describes the base case, emphasizing that it must be proven for a specific initial value*. D. The statement is assumed true for n+1. This is part of the inductive step, where we assume P(k) is true to prove P(k+1). Option C provides the most accurate and complete description of the base case in mathematical induction. The correct option is C. C