Exercise 2.3.22* (Condition of solvability for the preceding system)

Proof.

$\text{(⇐)}$

Suppose that $a_i\equiv a_j(\mathit{mod}{p}^{{\alpha }_i})$ for all pairs of indices $i,j$ for which $1\le i<j<t$. In particular, for $j=r$, $$a_i\equiv a_r(\mathit{mod}{p}^{{\alpha }_i}),1\le i\le r.$$

By Problem 21, the system (2.1)

$$x\equiv a_1(\mathit{mod}{p}^{{\alpha }_1}),x\equiv a_2(mod{p}^{{\alpha }_2}),\cdots ,x\equiv a_r(mod{p}^{{\alpha }_r})(2.1)$$

has a simultaneous solution.

$\text{(⇒)}$

Suppose now that the system (2.1) has a simultaneous solution $x$. A fortiori, for every $j\in [[2,r]]$, $x$ satisfies the shorter system $$x\equiv a_1(\mathit{mod}{p}^{{\alpha }_1}),x\equiv a_2(mod{p}^{{\alpha }_2}),\cdots ,x\equiv a_j(mod{p}^{{\alpha }_j}).$$

By Problem 21, where we take $r=j$, we obtain $a_i\equiv a_j(\mathit{mod}{p}^{{\alpha }^i})$ for $i=1,2,\dots ,j$.

So $a_i\equiv a_j(\mathit{mod}{p}^{{\alpha }_i})$ for all pairs of indices $i,j$ for which $1\le i<j<t$.