How many positive integers 'n' are there such that [(2)^n + 1] is divisible by 7 ?
(In reply to
No Subject by K Sengupta)
At the outset, we know that the quadratic residues of mod 7 are 0,1,2,4
Now, we observe that: 2^1=2(mod7), 2^2=4(mod 7), 2^3=1( mod 7), 2^4=2(mod7), 2^5=4( mod 7), 2^6=1(mod7), and, so on......
Accordingly, the residues of 2^n in mod 7 system cycles as (2,4,1,2,4,1,.....)
=> The residues of 2^n +1 cycles as (3,5,2,3,5,2,.....)
Since none amongst 3, 5 and 2 correspond to a quadratic residue in mod 7 system, it follows that:
There DOES NOT EXIST any integer satisfying the given conditions.
Puzzle Solution
