Question

Let S be the following statement. A necessary condition for an integer to be divisible by 6 is that it be divisible by 3.

(a) Which of the following is another way to express Statement S?
i. If an integer is divisible by 3, then it is also divisible by 6.
ii. If an integer is divisible by 6, then it is also divisible by 3.
(b) Is Statement Strue or false? Answer the question by selecting sentences from the following scrambled list and putting them in the correct order.
1. Then n = or for some integer r, and so n = 3(2r).
2. Then 6 = 3r for some integer r.
3. Counterexample: Let n = 15.
4. Proof: Suppose n is any integer that is divisible by 3.
5. Then n = 3r for some integer r.
6. This proof shows that the statement is true.
7. Since 2r is a product of integers, it is an integer, and, hence, n is divisible by 3.
8. This counterexample shows that the statement is false.
9. Proof: Suppose n is any integer that is divisible by 6.
10. Then n is divisible by 3 but n is not divisible by 6.

Answer 1

Statement S correctly expresses that a necessary condition for an integer to be divisible by 6 is that it also be divisible by 3. It is confirmed true with a proof sequence of statements: 4, 5, 1, 7, 6. The sentences from the scrambled list provide a logical demonstration that for integers divisible by 3, being divisible by 6 follows logically.

Step-by-step explanation:

The question relates to the concept of necessary and sufficient conditions in mathematical logic, specifically dealing with integers and their divisibility properties. A necessary condition for an integer to be divisible by 6 is that it be divisible by 3. Therefore:

Statement S is another way of expressing that if an integer is divisible by 6, then it must also be divisible by 3, which corresponds to (ii).

To assess if Statement S is true or false, we can organize the given sentences in a logical sequence as follows: 4, 5, 1, 7, 6. This sequence develops a proof showing that for any integer divisible by 3, being a multiple of 6 is a plausible result, hence the statement is true. Because if an integer n is divisible by 3, then by definition, n = 3r for some integer r (5), and multiplying by another integer 2 we have n = 3(2r) (1), which is the definition of being divisible by 6.

The false statements from the scrambled list, such as the counterexamples provided, do not apply since they suggest that an integer could be divisible by 3 without being divisible by 6, which does not contradict the statement, as it only establishes a necessary condition, not a sufficient one.