Elementary Number Theory Problems 4.2 Solution (David M. Burton's 7th Edition) - Q14

My Solution for "Give an example to show that $a^{k} \equiv b^{k} \pmod {n}$ and $k \equiv j \pmod {n}$ need not imply that $a^{j} \equiv b^{j} \pmod {n}$."

Ran
Ran


Background

All theorems, corollaries, and definitions listed in the book's order:

Theorems and Corollaries in Elementary Number Theory
All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 4)

I will only use theorems or facts that are proved before this question. So, you will not see that I quote theorems or facts from the later chapters.

Question

Give an example to show that $a^{k} \equiv b^{k} \pmod {n}$ and $k \equiv j \pmod {n}$ need not imply that $a^{j} \equiv b^{j} \pmod {n}$.

Solution

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