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If n is a multiple of 2p, then n is a multiple of p. Which of the following is the contrapositive?

Rationale

If n is not a multiple of p then n is not a multiple of 2p.

The contrapositive of a statement maintains logical equivalence and is derived by negating both the hypothesis and conclusion and then swapping them. In this case, the original statement "If n is a multiple of 2p, then n is a multiple of p" leads us to the contrapositive: if n is not a multiple of p, then n cannot be a multiple of 2p.

A (If n is a multiple of p then n is a multiple of 2p) YOUR ANSWER

This statement incorrectly asserts that being a multiple of p guarantees being a multiple of 2p. This is not logically valid since there are multiples of p that are not multiples of 2p, such as p itself when p is odd.

B (If n is a multiple of p then n is not a multiple of 2p) YOUR ANSWER

This choice is also incorrect because it suggests that any number that is a multiple of p cannot be a multiple of 2p. This is false as there are many cases where n can be both a multiple of p and a multiple of 2p, especially when p is even.

C (If n is not a multiple of p then n is not a multiple of 2p) CORRECT

This is the correct contrapositive statement. If n cannot be expressed as a multiple of p, it logically follows that n cannot be expressed as a multiple of 2p, since 2p is a larger multiple that requires p to be a factor of n.

D (If n is not a multiple of 2p then n is not a multiple of p) YOUR ANSWER

This statement is misleading because it implies that failing to be a multiple of 2p also means failing to be a multiple of p. However, n could still be a multiple of p without being a multiple of 2p; for example, if p is 3 and n is 3.

Conclusion

The contrapositive of a conditional statement provides a crucial logical equivalence that helps in understanding implications. In this case, the correct contrapositive, "If n is not a multiple of p then n is not a multiple of 2p," accurately reflects the necessary relationship between multiples of p and 2p, reinforcing the foundational principle of logical reasoning in mathematics.

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