Is this number prime or composite?Prove your answer

Question

Is the number $1111...$($91$ times $1$) prime or composite? I have tried to break it into multiples of 10,but that just gave me a huge binomial expandum.

Answer 1

Hint:

The given number is $N=\frac{10^{91}-1}{9}$ is a repunit. Note that $10^{91}-1=(10^{13})^7-1=(10^{13}-1)(\text{some number }>1)$

Answer 2

11111...(91 x 1) is a repunit, which is an integer in which every digit is one.

Properties of repunits include:

Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition.

For example,

R35(b) = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001, since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base b in which the repunit is expressed.

R91 is a repunit with a composite number of digits (91)

Therefore, R91 is a composite number.