Factoring the repunit 111111

Text

A repunit is a number whose digits are all equal to $1$. Consider $R_6 = 111111$. Factor it completely into primes and state its largest prime factor.

Solution

It helps to split the number using two well-known factorizations: $111111 = 111\cdot 1001.$ Now $111 = 3\cdot 37$ and $1001 = 7\cdot 11\cdot 13$. Therefore $111111 = 3\cdot 7\cdot 11\cdot 13\cdot 37,$ and the largest prime factor is $\boxed{37}$.