# An alternative factorization

**Number Theory**Level 5

\[\large\displaystyle { x }^{ 100 }-1\equiv \prod _{ i> 0 }^{ \quad }{ (x+{ a }_{ i }) } \pmod{101}\]

For natural numbers \(a_i\), consider the congruence above. Observe that these are linear factors, so a standard factorization for integers will not work (in fact you'll have a 50th power in there). Determine the minimum sum of all \({a}_{i}\).