Question: I'm attempring to solve a congruence $x^2 \equiv 17\pmod{128}$ but not quite sure how to go about it. I see that $128 = 2^7$, but the Chinese Remainder Theorem doesn't apply to $\gcd > 1$. I found one solution quite easily by finding solutiong to $x^2 \equiv\pmod{32}$ which was $x \equiv 23\pmod{32}$ and it turned out that $23 + 128x\_0 \equiv 17\pmod{128}$. How do I find the other solution ans what's the proper way of doing it?

Answer: 
$x^2\equiv17\pmod{128}\implies x^2\equiv17\pmod{16}\equiv1$

$(4a+1)^2\equiv8a+1\pmod{16},$ 

we need $8a+1\equiv17\pmod{16}\implies a$ must be even $=2b$(say)

Again, $x^2\equiv17\pmod{128}\implies x^2\equiv17\pmod{32}$

Now $(8b+1)^2\equiv16b+1\pmod{32},$ we need $16b+1\equiv17\pmod{32}\implies b=2c+1$

$8b+1=8(2c+1)+1=16c+9$

Again, $x^2\equiv17\pmod{128}\implies x^2\equiv17\pmod{64}$

and $(16c+9)^2\equiv288c+81\pmod{64}\equiv32c+17$

For $(16c+9)^2\equiv17\pmod{64},c=2d$

$16c+9=16(2d)+9=32d+9$

Now $(32d+9)^2\equiv576d+81\equiv64d+81\pmod{128}$

We need $\displaystyle64d+81\equiv17\pmod{128}$
$\iff64d\equiv-64\pmod{128}\iff d\equiv-1\pmod2\implies d=2e-1$

$32d+9=32(2r-1)+9=64r-23\equiv-23\pmod{64}$

Similarly start with $4a-1$ to get $x\equiv23\pmod{64}$

That these are the two exclusive in-congruent solutions can be verified as follows:

$$x^2\equiv17\pmod{128}\equiv23^2$$

$$\iff128|(x-23)(x+23)$$

$$\iff32\mid\dfrac{x-23}2\cdot\dfrac{x+23}2$$

As $\dfrac{x+23}2-\dfrac{x-23}2=23$ is odd, so, they must be of opposite parity

If $32\mid\dfrac{x-23}2,x\equiv23\pmod{64}$

and $\cdots$
Let $x$ and $y$ be relatively prime with $128$. If $x^2=y^2$ then $(xy^{-1})^2=1$ so $xy^{-1}=\pm1 \iff x=\pm y$, this tells us the number of solutions is two or zero, lets look for the solutions in this case.

It is a somewhat well-know result that $5$ generates half of $\mathbb Z\_{2^n}^{\cdot}$. So lets look at the powers of $5\bmod 128$:

$1,
5,
25,
125,
113,
53,
9,
45,
97,
101,
121,
93,
81,
21,
105,
13,
65,
69,
89,
61,
49,
117,
73,
109,
33,
37,
57,
29,
17$

Oh, from the looks of it $5^{28}\equiv 17\bmod 128$. Therefore $(5^{14})^2\equiv 17\bmod 128$.

So our solutions are $5^{14}$ and $-(5^{14})$, according to our previous calculations $5^{14}\equiv 105$. So the solutions are $105$ and $23$