Mathematics, 08.03.2021 22:30 heavenmcgautha
For each positive integer n, consider the pair (3^n β1, 5^n β1). 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 β 1, 5^n β 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 β 1 and 5n β 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.
Answers: 3
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For each positive integer n, consider the pair (3^n β1, 5^n β1). For n = 1, this is the pair (2, 4)...
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