exist $x$ such that $x^k \equiv m$ mod $(p_1\cdot p_2) \Leftrightarrow $ exists $x_1,x_2$ : $x_1^k\equiv m(p_1)$ and $x_2^k\equiv m(p_2)$

Question

Let $p_1,p_2$ prime numbers, I wish to show that: exist $x$ such that $x^k \equiv m$ mod $(p_1\cdot p_2) \Leftrightarrow $ exists $x_1,x_2$ : $x_1^k\equiv m(p_1)$ and $x_2^k\equiv m(p_2)$

A first approach I took, was to use $y\equiv m(p_1) , y\equiv m(p_2) \Leftrightarrow y\equiv m(p_1\cdot p_2)$, then by assigning $x^k=y$ the problem comes to find whether $y$ has a $k$-order root in $U(\mathbb{Z}_{p_1\cdot p_2})$. How ever it doesn't seem to simplify the problem.

A second approach I took was to use the fact which derived from CRT , that $U(\mathbb{Z}_{p_1 \cdot p_2}) \cong U(\mathbb{Z}_{p_1}) \times U(\mathbb{Z}_{p_2}) $, In $U(z_{p_i})$ which are cyclic groups, there is a solution for $x^k \equiv m(p_i) \Leftrightarrow m^{\frac{p_1-1}{gcd(k,p_1)}}=1 (p_i)$. So assuming $gcd(k,p_1) = 1$ there are solutions for the equations $x_1,x_2$. But I am struggling to show that $\pi^{-1}(x_1,x_2)$ (when $\pi$ is the isomorphism from CRT), is a solution for $x^k \equiv m (p_1p_2)$.

So in case my second approach is correct, I would be glad for some help with showing $\pi^{-1}(x_1,x_2)$ is a solution, and also in case $x$ is a solution mod $(p_1p_2)$ then $\pi_1(x) ,\pi_2(x)$ are solutions mod $p_1$, $p_2$ respectively.

Also other approaches or ideas would be appreciated.

related question:

If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$

Answer 1

$x ≡a\mod (p_1)$ , if $x=(kp_2)p_1+a$

$x ≡a\mod (p_2)$ , if $x=(lp_1)p_2+a$

Then:

$2x=(k+l)p_1p_2+2a$

If $2|k+l$, then:

$x=(\frac{k+l}{2})p_1p_2+a$

Or; $x ≡a\mod (p_1)(p_2)$

Example:

$72=2.5.7+2 ≡2\mod 5 ≡2\mod 7 ≡2\mod 35 $

We may assume $kp_1=x_1$ and $lp_2=x_2$ and we have:

$x_1^t=k^t.p_1^t$

$x_2^t=l^t.p_2^t$

If $l^t.p_2^{t-1}=k^t.p_1^{t-1}=m$, then:

$x_1 ≡m\mod (p_1)$

$x_2 ≡m\mod (p_2)$