Elementary Number Theory Problems 3.2 Solution (David M. Burton's 7th Edition) - Q14

Background

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

Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)

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 - 3)

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 previous problem to obtain the prime factors of the repunit $R_{10}$.

Solution

From Q13(a), we know "If $n \mid m$, then $R_{n} \mid R_{m}$". Thus $2 \mid 10$ implies that $R_{2} \mid R_{10}$. $R_{2} = 11$ which is a prime. Similarly,