Example for $a^k\equiv b^k$ and $k\equiv j$ but $a^j\not\equiv b^j\pmod n$

Question

I need some help in the number theory please , Who can give me an example : If $$a^k≡b^k \pmod{n}$$ and $$k≡j \pmod{n}$$ is not necessary to be $$a^j≡b^j \pmod{n}$$

Answer 1

One counter example to you question is : $$1^2\equiv 2^2\mod 3 ,\text{ and } 5\equiv 2\mod 3 \text{ and } 2^5 \equiv 2 \not \equiv 1^5\mod 3$$ Note that if we modify a little your statement, if$a,b$ are coprime then we have : $$\left(a^k\equiv b^k\mod n\ \text{ and }\ \ k\equiv j\mod \varphi(n)\right)\Rightarrow a^j\equiv b^j\mod n $$ with $\varphi(n)$ is the Euler's totient function