Ex 9.1, 8

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Ex 9.1, 8 Frame the converse for each of the propositions in Questions 1–12. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement. In Questions 8–12, n is a positive integer If n is divisible by 24, then it is divisible by both 4 and 6. Let’s write Proposition and Converse first Proposition: If n is divisible by 24, then it is divisible by both 4 and 6. Converse: If n is divisible by both 4 and 6, then it is divisible by 24. Checking if Proposition is true Proposition: If n is divisible by 24, then it is divisible by both 4 and 6 Let’s prove this Since n is divisible by 24, we can write 𝒏=𝟐𝟒𝒌 for an integer 𝑘. Now, we can write 𝑛=24𝑘=𝟒 × (𝟔𝒌) This means n is divisible by 4 And: 𝑛=24𝑘=𝟔 × (𝟒𝒌) This means n is divisible by 6 Thus 𝑛 is divisible by both 4 and 6 . ∴ Proposition is true Checking if Converse is true Converse: If n is divisible by both 4 and 6, then it is divisible by 24. Let’s prove this Let n = 12 Here, 12 is divisible by 4 12 is divisible by 6 12 is NOT divisible by 24 ∴ Converse is false