Prove the following statement by mathematical induction. For every integer n 2 0, 7" - 1 is divisible by 6. Proof (by mathematical induction): Let P(n) be the following sentence. 7n - 1 is divisible by 6. We will show that P(n) is true for every integer n 2 0. Show that P(0) is true: Select P(0) from the choices below. O (7° - 1) | 6 O 61 (7° – 1) O 1 is a factor of 7° - 1 O 6 is a multiple of 70 - 1 The truth of the selected statement follows from the definition of divisibility and the fact that 7° - 1 = Show that for each integer k 2 0, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 0, and suppose that P(k) is true. Select P(k) from the choices below. O 6 is a multiple of 7k - 1 1 is a factor of 7* - 1 O 6 is divisible by (7* - 1) O (7k - 1) is divisible by 6 [This is P(k), the inductive hypothesis.] We must show that P(k + 1) is true. Select P(k + 1) from the choices below. O (7k * 1 - 1) is divisible by 6 O 1 is a factor of 7k + 1 _ 1 6 is a multiple of 7k *1 - 1 O 6 is divisible by (7k +1 - 1) By the inductive hypothesis and the definition of divisibility, there exists an integer r such that 7k - 1 = 6r, and so 7k - 6r+ 1. Now 7k+1 - 1 = 7*.7 - 1. When 6r + 1 is substituted for 7k in the above equation, and the right-hand side is simplified, the result can be expressed in terms of k and r as follows. zk+1 - 1 = 6· This quantity is an integer because k and r are integers. Select the final sentence from the choices below. Hence, 6 is a multiple of (7**1 - 1), and so P(k+1) is false, which completes the inductive step. O Hence, 6 is divisible by (7**1 - 1), and so P(k+1) is true, which completes the inductive step. O Hence, (7k+1 1) is divisible by 6, and so P(k+1) is false, which completes the inductive step. Hence, (7**1 - 1) is divisible by 6, and so P(k+1) is true, which completes the inductive step. O Hence, 1 is a factor of (7**1 - 1), and so P(k+1) is false, which completes the inductive step. O Hence, 1 is a factor of (7k+1 - 1), and so P(k+1) is true, which completes the inductive step.