Use the theory of congruences to verify that 89 | 2^{44} - 1 and 97

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)

RanblogRan

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

Use the theory of congruences to verify that $$89 \mid 2^{44} - 1 \qquad \text{and} \qquad 97 \mid 2 ^{48} - 1$$

Solution

For $89 \mid 2^{44} - 1$:

$$ \begin{equation} \begin{split} 2^{44} - 1 & \equiv (2^{11})^{4} - 1 \\ & \equiv 2048^{4} - 1 \\ & \equiv 1^{4} - 1 \\ & \equiv 0 \pmod {89} \end{split} \nonumber \end{equation} $$

So, $89 \mid 2^{44} - 1$.