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Sum of factorials divisible by p

Source: Simon Marais MC 2019 B2

October 14, 2019
factorialnumber theory

Problem Statement

For each odd prime number pp, prove that the integer 1!+2!+3!+⋯+p!−⌊(p−1)!e⌋1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rflooris divisible by pp (Here, ee denotes the base of the natural logarithm and ⌊x⌋\lfloor x\rfloor denotes the largest integer that is less than or equal to xx.)
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