# Is there a nice factorization?

**Number Theory**Level 4

\[256+128\left(\sum_{1\leq i\leq 8} a_i\right) + 64\left(\sum_{1\le i<j\le 8} a_ia_j\right) + \\ + 32\left(\sum_{1\le i<j<k\le 8} a_ia_ja_k\right) + \ldots + a_1a_2\cdots a_7a_8 = 111546435 \]

There exist some positive integers \(a_{i}\) for \(i=1,2,..8\) such that they satisfy the above equation find the sum of all possible value of \(a_{3}\)

*hint - 111546435 has quite interesting factors.*