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