solve Equations with Modulus(%)

Question

I have been working on this for a while now and not seem to solve this equation. The part that tripping me off is the mod. How do I take it to the other side so I can solve for $x$.

Suppose: $X = 10, a=20, z=30$, how to solve for smaller $x$. $X=a^x \bmod z$

I tried using logs but answer is wrong. Could someone please guide me on how to do this?

Answer 1

Hint: Calculating the first powers of $20$ mod.$30$, you can conjecture (and prove) that $$20^n\equiv 20\quad\text{if $n$ is odd}, \qquad 20^n\equiv 10\quad\text{if $n$ is even.} $$

Answer 2

$x$ is even is a solution. $10\equiv20^x(mod30)$

10,20,30 all are divisible by 10 $10\equiv 1(mod3)$

$20^x\equiv2^x(mod 3)$

But 30 is a multiple of 3

Therefore $20^x\equiv1(mod 3)$ This is possible for $x$ is even