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If $a, \ b, \ c, \ d$ are integers and $a \equiv b \pmod c$ then $d^a \equiv d^b \pmod c$. True or false?

I changed this statement to If $a,b,c,d$ are integers and $c\mid (a-b)$ then $c\mid (d^a - d^b)$ for the sake of my understanding.

Any hints on how this statement can be proven or disproven?

Anonymous Computer
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baba
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    The following is true: if $\gcd(d, c) = 1$ and $\varphi$ is the Euler totient function, then $$a \equiv b \mod \varphi(c) \implies d^a \equiv d^b \bmod c $$ –  Jun 05 '14 at 02:11

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It is false. Take $c=3, a=5,b=2,d=-1$ for a counterexample.

please delete me
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  • is there a systematic way to prove this? or is proof by counterexample the only way – baba Jun 05 '14 at 02:09
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    @baba: This is one of those problems where just about every example is a counter-example, so all you need for a systematic approach is "pick something, check it doesn't work". –  Jun 05 '14 at 02:12