Algebra - congruence relations

Question

I suppose its not too complicated but I could find the answer or answer myself ....

Here is the problem:

$x$ is a whole number

$x<m$

$x^{p_1} \equiv r_1\pmod{m}$

$x^{p_2} \equiv r_2 \pmod{m}$

find $x$

For example:

$x$ is a whole number

$x<33$

$x^{7} \equiv 29\pmod{33}$

$x^{13} \equiv 8\pmod{33}$

find $x$

Answer 1

Note that $7$ and $13$ are relatively prime, so there exists an expression of the form $7a + 13b = 1$. Specifically $7\cdot 2 - 13 = 1$.

Now $(x^7)^2(x^{13})^{-1} = x^{7\cdot 2 - 13} = x \pmod{33}$ so $x = 29^2\cdot8^{-1} \pmod{33}$. And here $8^{-1}$ doesn't mean $\frac18$, it means the number $0 \leq y < 33$ such that $8y \equiv 1 \pmod{33}$.

Do you know how to compute $8^{-1}$? If you do then you're done.