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N ≡ 1 ! + 2 ! + 3 ! + ⋯ + 2 0 1 9 ! (mod 100) ≡ 1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! + 7 ! + 8 ! + 9 ! (mod 100) ≡ 1 + 2 + 6 + 2 4 + 2 0 + 2 0 + 4 0 + 2 0 + 8 0 (mod 100) ≡ 1 + 2 + 6 + 4 (mod 100) ≡ 1 3 (mod 100) Since n ! m o d 1 0 0 = 0 for n ≥ 1 0