When is 2 to the n plus 1 divisible by 3?

Question

Look at 2 to the n plus 1 for n = 1, 2, 3, 4, 5, 6: the values are 3, 5, 9, 17, 33, 65.

For which values of n is 2 to the n plus 1 divisible by 3? Prove it. Then answer the same question for 2 to the n minus 1.

Accepted Answer

2 to the n plus 1 is divisible by 3 exactly when n is odd. And 2 to the n minus 1 is divisible by 3 exactly when n is even. The remainder argument. The number 2 leaves remainder 2 when divided by 3, and 2 is the same as minus 1 modulo 3. So 2 to the n behaves like (minus 1) to the n modulo 3: it leaves remainder 2 (that is, minus 1) when n is odd and remainder 1 when n is even. - n odd: 2 to the n is minus 1 modulo 3, so 2 to the n + 1 is 0 modulo 3: divisible. - n even: 2 to the n is 1 modulo 3, so 2 to the n + 1 is 2 modulo 3: not divisible, while 2 to the n minus 1 is 0 modulo 3: divisible…

When is 2 to the n plus 1 divisible by 3?

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