Remainder of factorials

Number Theory Level 2

What is the reminder when 1 ! + 2 ! + 3 ! + ⋯ + 5670 ! 1! + 2! + 3! + \cdots + 5670! is divided by 24 24 ?


The answer is 9.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

24 = 4 ! ∣ n ! 24=4!|n! for any positive integers n ≥ 4 n\ge4 . Therefore, the remainder is 3 ! + 2 ! + 1 ! = 9 3!+2!+1!=9

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