Factor It 2

The number $8888888899999999$ consists of $8 \textrm{ }8s$ and $8 \textrm{ } 9s$ , making it divisible by $11$ based on the divisibility rule. This results in $808080809090909=1010101\cdot 9 + 1010101\cdot 8 \cdot 10^8=1010101\cdot (9+8000000000)=(101 \cdot (1+10^4) \cdot (8000000009) = 101 \cdot 10001 \cdot 8000000009$ From there, I wrote a program to find the prime factors of the remaining terms and got $10001=73\cdot 137$ and $8000000009=15299\cdot 52291$ . In these prime factors, $52291$ is the biggest, thus, the answer is $52291$

Prime factorization of 8888888899999999 is:

$11 \times 73 \times 101 \times 137 \times 15299 \times 52291$

Largest prime factor is $\boxed{52291}$

$8888888899999999 = 11111111 × 800000009 =$

$= ( 11×73×101×137 ) × (15299 × 52291)$

Hence, our answer should be:

$\boxed {52291}$