Question

Is the following statement true or false? for every integer n, if n is prime then (−1)n = −1.

a. True
b. False

Answer 1

The statement is true. For every integer n, if n is prime, then (-1)^n = -1.

Step-by-step explanation:

The statement is true.

This is because if n is prime, then it cannot be divided evenly by any other integer except for 1 and itself. Since all integers can be expressed in the form of either an odd or an even number, let's consider both cases:

1. Odd integers: When n is an odd prime, we can write it as n = 2k+1, where k is an integer. In this case, (-1)k would be either 1 or -1. Thus, (-1)n = (-1)2k+1 = (-1)2k * (-1)1 = 1 * (-1) = -1.

2. Even integers: When n is an even prime, we can write it as n = 2k, where k is an integer. In this case, (-1)2k would always be 1, as any even exponent of -1 is equal to 1. Thus, (-1)n = (-1)2k = 1.