B2

Part of 2019 Simon Marais Mathematical Competition

Problems(1)

Sum of factorials divisible by p

Source: Simon Marais MC 2019 B2

For each odd prime number

p

, prove that the integer

1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor

is divisible by

p

e

denotes the base of the natural logarithm and

\lfloor x\rfloor

denotes the largest integer that is less than or equal to

x

factorialnumber theory