This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To prove that 7^n-1 is divisible by 7, we need to show that 7^n-1 = 7k for some integer k. We will consider the domain of n as positive integers (n \1, 2, 3, \). Step 1: Consider the case for n=1. Substitute n=1 into the expression: 7^1-1 = 7^0 = 1 For 1 to be divisible by 7, 1 must be equal to 7k for some integer k. However, 1/7 is not an integer. Therefore, 1 is not divisible by 7. This means the statement "7^n-1 is divisible by 7" is false for n=1. Step 2: Prove for integers n 2. For any integer n 2, the exponent n-1 will be a positive integer. Let m = n-1. Since n 2, we have m 1. The expression becomes 7^m. We can rewrite 7^m as: 7^m = 7 × 7^m-1 Since m 1, the exponent m-1 is a non-negative integer (m-1 0). Therefore, 7^m-1 is an integer. Let k = 7^m-1. Since 7^m-1 is an integer, k is an integer. So, we have: 7^m = 7k This shows that 7^m is a multiple of 7, and thus divisible by 7. Substituting back m = n-1, we conclude that 7^n-1 is divisible by 7 for all integers n 2. Conclusion: The statement "7^n-1 is divisible by 7" is not true for all positive integers n. It is false for n=1. However, it is true for all integers n 2.