$2^x$ exponential congruence

Question

How does one go about solving this congruence: $$2^x+8\equiv0\pmod{27}$$ I wrote a program to go over many values of x, and it seems that $x=12+18t$ where $t\in \Bbb N$. How would I get to that conclusion without going through every x? There was a similar question here: Solution to exponential congruence, but unless I misunderstood, these solutions only work if 27 was a prime.

score 1 · Answer 1 · answered Mar 07 '17 at 01:22

1

Use Euler's totient theorem with $\varphi(27) = 18$.

answered Mar 07 '17 at 01:22

Robert Israel

448,999

score 1 · Accepted Answer · answered Mar 07 '17 at 01:28

1

Tip: Find a generator of the group, write the equation as

$a^x=a^t \space mod \space k$

And solve $x=t\space mod\space \phi(k) \space\space\space$ $\phi$ representing Euler totient function

answered Mar 07 '17 at 01:28

Rab

1,176

$2^x$ exponential congruence

2 Answers2