Is $\underbrace{11111\ldots1}_{91\text{ times}}$ a prime number?

Question

Prove that $$\underbrace{11111\ldots1}_{91\text{ times}}$$ is a composite number and not a prime. Please give full steps of proving.

I tried and found that it is divisible by $1111111$ and $1111111111111$ but I can't prove it.

Answer 1

Since $7|91$ we see that we can write $1111111$-($7\;$ $1$'s). Next to itself $13$ times and get the result of $111...$ (-$91$ times.)
This is equivalent to saying that $$1111111+(10^7\times 1111111) + (10^{14}\times 1111111) + ... + (10^{84}\times 1111111) = 1111... -91 \text{times}$$.

Which we can also see using a geometric series. Taking a factor of $1111111$ out of the LHS gives your result that $111111$ divides $1111...$(-$91$ times)

Answer 2

$91=7\times13$. So the number is $\sum_{k=0}^{12}(1111111\times10^{7k})=1111111\sum_{k=0}^{12}10^{7k}$.