For each positive integer n, consider the pair (3^n −1, 5^n −1). For n = 1, this is the pair (2, 4) and for n = 2, it is (8, 24). Can you find a value for n such that both numbers in the pair (3^n − 1, 5^n − 1) are divisible by d = 7? Can you find more than one such n? Do you think there are finitely or infinitely many n such that 3n − 1 and 5n − 1 are both divisible by d = 7? Why? Explain your reasoning as carefully as you can. Now, what if we replace d = 7 with d = 11? Or with d = 13? What if we take d = 77
or d = 1001? Describe any patterns you think are interesting.