Question 46

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

The contrapositive of a statement is formed by negating both the hypothesis and the conclusion. In this case, the contrapositive of "If n is a multiple of 2p, then n is a multiple of p" is "If n is not a multiple of p, then n is not a multiple of 2p."

A) If n is a multiple of p then n is a multiple of 2p

This statement is not the contrapositive; rather, it incorrectly reverses the original proposition. It implies that being a multiple of p guarantees being a multiple of 2p, which is not necessarily true.

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

This option is also incorrect as it suggests that being a multiple of p excludes the possibility of being a multiple of 2p. This contradicts the original statement, which allows for the possibility of n being a multiple of both p and 2p.

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

This is indeed the contrapositive of the original statement. It correctly states that if n does not satisfy the condition of being a multiple of p, then it cannot satisfy the condition of being a multiple of 2p, aligning perfectly with logical reasoning.

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

While this statement might seem related, it is not the contrapositive of the original proposition. Instead, it suggests that failing to be a multiple of 2p implies failing to be a multiple of p, which does not logically follow from the original statement.

Conclusion

The contrapositive of a conditional statement is logically equivalent to the original statement, and in this case, option C accurately reflects that relationship. Options A, B, and D fail to capture the necessary logical structure, making C the only correct choice.