# An alternative factorization

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

For positive integers $a_i,$ 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 $50^\text{th}$ power in there$).$ Determine the minimum sum of all ${a}_{i}.$

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