How to tell if system of congruences where each base is a power of prime $p$ has a solution

Question

$p$ is a prime number.

How to tell if a system of congruences:

\begin{align} x &\equiv a_1 \pmod{p^{i_1}} \\ x&\equiv a_2 \pmod{p^{i_2}} \\ &\dots\\ x &\equiv a_n \pmod{p^{i_n}} \end{align}

Has a solution.

How would you find the solution?

I feel like it has something to do with the chinese remainder theorem but the mod bases are definitely not pairwise relatively prime

Answer 1

If I understand your question correctly, you have \begin{align*} x & \equiv a_1 \pmod{p^{e_1}}\\ x & \equiv a_2 \pmod{p^{e_2}}\\ \vdots & \equiv \vdots\\ x & \equiv a_k \pmod{p^{e_k}} \end{align*}

Without the loss of generality, let us assume $e_1 \leq e_2 \leq \dotsb \leq e_k$. If the last congruence $x \equiv a_k \pmod{p^{e_k}}$ has a solution, then we should have $x \equiv a_k \pmod{p^j}$ for all $j \leq e_k$. In other words, all the congruences with smaller powers should also give the same residue. So you need $a_1 \equiv a_2 \equiv \dotsb$ for the solution to exist.

Answer 2

The general condition for a system of congruences with non-necessarily coprime moduli $$x\equiv a_i\pmod{m_i}\qquad (1\le i\le r)$$ is that $$\forall (i,j), 1\le i,j\le r,\enspace a_i\equiv a_j\mod \gcd(m_i,m_j).$$

Answer 3

In general, $x \equiv a \pmod{p^\alpha}$ and $x \equiv b \pmod{p^\beta}$ if and only if $a \equiv b \pmod{p^\beta}$.

For example, $x \equiv a \pmod{13^5}$ and $x \equiv b \pmod{13^3}$ if and only if $a \equiv b \pmod{13^3}$