How many positive integers 'n' are there such that [(2)^n + 1] is divisible by 7 ?

This problem is easily solved with mod 7 arithmetic (ie Base 7 arithmetic - usually learned in number theory).

We are looking for solutions for n such that 2^n = 6 mod 7.

2^1 = 2 mod 7

2^2 = 4 mod 7

2^3 = 1 mod 7

2^4 = 2*(2^3) mod 7 = 2 mod 7

2^5 = 2*(2^4) mod 7 = 4 mod 7

2^6 = 2*(2^5) mod 7 = 1 mod 7

etc.

There is no solution for n.