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 $${a}_{i}$$.

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