If a,b,c,k $\in\mathbb{Z\cap{(N\cup{0}})}$ and a$\equiv$b(mod c) then prove that a$^{k}\equiv$b$^{k}$(mod c). I know how to prove it using induction but I wanted to know if there is a method that only uses number theory.
2026-03-25 04:35:07.1774413307
Modulo Arithmetic: Proof of Basic Property
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If $a \equiv b \pmod{c}$, then these two guys are the same element of the ring of integers modulo $c$. Hence the $k$-th power of one is the same as the $k$-th power of the other.
For this you need to prove that the product in the quotient ring is well defined (no induction) and then you can conclude (with induction, if you really want to be that formal) that multiplying $k$ times one is the same as multiplying $k$ times the other one.