Question

Disprove the following statement by giving a counterexample. For every integer n, if n is even then n2 + 1 is prime.
Counterexample: Consider the ordered pair (n, n2 + 1) = ( )
The values in the ordered pair show that the given statement is false because (choose one)
a. n is even and n2 + 1 is prime.
b. n is even and n2 + 1 is not prime.
c. n is not even and n2 + 1 is prime.
d. n is not even and n2 + 1 is not prime.

Answer 1

All the statements are incorrect.

Option a is a false statement. If n is even, then n2 + 1 might not be a prime number. e.g. let n is 8, then n² be 64, and n² + 1 is 65, which is not a prime number.

Option b is a false statement. If n is even, then n2 + 1 might be a prime number. e.g. let n is 2, then n² be 4, and n² + 1 is 5, which is a prime number.

Option c is a false statement. If n is not even i.e. odd, then n² + 1 might not be a prime number. e.g. let n is 3, then n² be 9, and n² + 1 is 10, which is not a prime number.

Option d is a false statement. If n is not even i.e. odd, then n² + 1 might be a prime number. e.g. let n is 1, then n² be 1, and n² + 1 is 2, which is a prime number.

So, all the statements are incorrect.

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