Counter-Examples

Some questions ask you to find a counter-example to a given statement. This means that you must find an example which renders the conclusion of the statement false. If you must select a counter-example among multiple choices, often you can use the trial and error approach to determine which of those choices leads to a contradiction.

Other questions are more open-ended and require you to think more creatively. Common values that lead to contradictions are -1, 0, 1, and 2, but each problem has unique givens and restrictions, and you must keep those in mind. Also, there isn't a counter-example to a true statement. If you find yourself testing value after value to no avail, you should consider proving the statement true.

Finding a counter-example to each answer choice may be the fastest way to solve the problem. Remember, one counter-example to a statement is enough to disprove it.

Which of the following numbers is a counter-example to the following claim?

If $p$ is an odd prime, then $p + 2$ is also a prime.

(A) $\ \ 3$
(B) $\ \ 5$
(C) $\ \ 7$
(D) $\ \ 9$
(E) $\ \ 11$

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Correct Answer: C

Solution:

Because we are looking for a counter-example, we must find an odd prime number $p,$ such that $p + 2$ is NOT a prime. We analyze each of the choices.

(A) 3 is a prime, and $3 + 2 = 5$ is a prime. This is not a counter-example.
(B) 5 is a prime, and $5 + 2 = 7$ is a prime. This is not a counter-example.
(C) 7 is a prime, and $7 + 2 = 9$ is NOT a prime. This is a counter-example.
(D) 9 is a not a prime. This is not a counter-example.
(E) 11 is a prime, and $11 + 2 = 13$ is a prime. This is not a counter-example.

Hence, the answer is (C).

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Incorrect Choices:

(A), (B), (D), and (E)
The solution explains how to eliminate these choices.

If you got this problem wrong, you should review Prime Numbers.

Which of the following numbers is a counter-example to the following claim?

If $n$ is an integer, then $n^2 + 1$ is a prime.

(A) $\ \ 1$
(B) $\ \ 2$
(C) $\ \ 3$
(D) $\ \ 4$
(E) $\ \ 3 \sqrt{2}$

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Correct Answer: C

Solution:

Because we are looking for a counter-example, we must find an integer $n$ such that $n^2 + 1$ is NOT a prime. We analyze each of the choices.

(A) 1 is an integer, and $1^2 + 1 = 2,$ which is a prime. This is not a counter-example.
(B) 2 is an integer, and $2^2 + 1 = 5,$ which is a prime. This is not a counter-example.
(C) 3 is an integer, and $3^2 + 1 = 10 = 2 \times 5,$ which is NOT a prime. This is a counter-example.
(D) 4 is an integer, and $4^2 + 1 = 17,$ which is a prime. This is not a counter-example.
(E) $3 \sqrt{2}$ is not an integer. This is not a counter-example.

Thus, the answer is (C).

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Incorrect Choices:

(A), (B), (D), and (E)
The solution explains how to eliminate these choices.

If you got this problem wrong, you should review Prime Numbers.

[[start-example]]

Which of the following cannot be the perimeter of a triangle whose side lengths are all integers?

(A) $\ \ 2$
(B) $\ \ 3$
(C) $\ \ 5$
(D) $\ \ 7$
(E) $\ \ 11$

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Correct Answer: A

Solution:

Let's consider each choice.

(A) If the side lengths are positive integers, then they are at least 1, and hence the perimeter cannot be 2. Thus, this choice is not possible.
(B) The triangle with side lengths $1-1-1$ has a perimeter of 3.
(C) The triangle with side lengths $2-2-1$ has a perimeter of 5.
(D) The triangle with side lengths $3-3-1$ has a perimeter of 7.
(E) The triangle with side lengths $5-5-1$ has a perimeter of 11.

Hence, the answer is (A).

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Incorrect Choices:

(B), (C), (D), and (E)
The solution explains how to eliminate these choices.

If you got this problem wrong, you should review Triangles.