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As a simple case take $k=2, a\equiv b\pmod {p^2}\implies a\equiv b\pmod p.$
A constructive example for $a= 2, b=27, p =5$ will work, but needed is a general or non-constructive approach, so as to get the reasoning applicable generally in a formal manner.

I mean the reason for the no change in residue class, for a root of the modulus is not clear.

I can only work out the case when $a , b \lt p$, as it is the trivial case (with $a=b$) for the smaller modulus case.

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closed as unclear what you're asking by Lord Shark the Unknown, Namaste, JMP, Trevor Gunn, Mohammad Riazi-Kermani Feb 16 '18 at 13:35

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    $\begingroup$ What could possibly be "non-constructive" about a proof of this (which just amounts to noting that $p\mid p^k$)? $\endgroup$ – Lord Shark the Unknown Feb 15 '18 at 7:07
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    $\begingroup$ I wonder what you think "non-construction proof" means. $\endgroup$ – fleablood Feb 15 '18 at 7:09
  • $\begingroup$ @LordSharktheUnknown For me your statement is a non-constructive proof, in a verbal manner. $\endgroup$ – jitender Feb 15 '18 at 7:09
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    $\begingroup$ This is trivial. If $p^2|a-b $ and $p|p^2$ then $p|a-b $. p doesn't have to be prime and we don't have to square it. If $a\equiv b \mod nk $ then $a\equiv b\mod n $. I'm pretty sure you prove that last week. $\endgroup$ – fleablood Feb 15 '18 at 7:13
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    $\begingroup$ I was talking more about a basic prove that if $a|b$ and $b|c $ then $a|c $. This question is absolutely the exact same. $\endgroup$ – fleablood Feb 15 '18 at 7:20
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$$a \equiv b \pmod{p^k}$$

$$a-b \equiv 0 \pmod{p^k}$$

$$\exists m \in \mathbb{Z}, (a-b) =p^km$$

$$\exists m \in \mathbb{Z}, (a-b) =p(p^{k-1}m)$$

$$a-b \equiv 0 \pmod{p}$$

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  • $\begingroup$ Thanks a lot. I request that the failure for the converse case be proved by such (non-constructive) approach. I mean that given $a\equiv b \pmod {p} \nrightarrow a \equiv b \pmod {p^k}$. $\endgroup$ – jitender Feb 15 '18 at 7:18
  • $\begingroup$ scratch head, to disprove it, I just need a counter example... I dont get what is non-constructive. $\endgroup$ – Siong Thye Goh Feb 15 '18 at 7:21
  • $\begingroup$ I mean that there should be an algebraic proof, but as in converse the hypothesis and conclusion are exchanged; and given the implication in the original - so, assuming hypothesis is H, conclusion is C; we have originally: $\lnot H \cup C$, and for converse, it is: $\lnot C \cup H$. $H: a\equiv b \pmod {p^k}$, $C: a\equiv \pmod p$. So, have: $\lnot (a\equiv b \pmod p) \cup H$. And need to prove it false, as a general case. Also, $\exists k \in \mathbb{Z}, a \equiv b \pmod p \implies (a-b) = kp$ $\endgroup$ – jitender Feb 15 '18 at 7:34
  • $\begingroup$ $p \equiv 0 \pmod{p}$ but $p \not \equiv 0 \pmod{p^k}, k>1$. $\endgroup$ – Siong Thye Goh Feb 15 '18 at 7:39
  • $\begingroup$ I hope yours is till a constructive example, just not using an actual value to substitute for. May be I am wrong, as if a general case is taken then it is non-constructive. Confused. But, one thing is certain that it is still showing a counter-case. $\endgroup$ – jitender Feb 15 '18 at 7:42
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If $a|b $ and $b|c $ then $a|c $. (Because if there exist $k $ so that $b=a*k $ and if there is a $m $ so that $c=b*m$, then $c=a (k*m) $

So if $a\equiv b\mod p^2$ that means $p^2|a-b $. And since $p|p^2$ we know $p|a-b $

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