What is the remainder when 1! + 2! + 3! + 4! + 5! .... + 50! is divided by 5! ?

Question

What is the remainder when 1! + 2! + 3! + 4! + 5! .... + 50! is divided by 5!   ?

zyberg · Answer

$ 1! + 2!+3!+ \cdots +50! \equiv  ? \pmod{5!}$

mjdennis · Answer

@ILovePuppies , you are using some of the notation wrong, I think.

ilovepuppies · Answer

Wait sorry

ilovepuppies · Answer

Nvm it's
1 + 2 + 3 + 50 (divided by) 5 = ?

ilovepuppies · Answer

Who's offline...

mjdennis · Answer

@Zyberg:
Wow.
Well, I'm going to trust that you are smart on the basics, that you understand what a factorial  is and how to use the distributive property.

First, distribute over the series
1!/5! + 2!/5! + ...  
Every part of the sum beyond 4! is evenly divisible by 5!, right?
5!/5! = 1
6!/5! = 6
7!/5! = 6*5
So none of these contribute to the remainder

Leaving you to solve ( 1! + 2! + 3! + 4! )

agent0smith · Answer

Very nice, @mjdennis

zyberg · Answer

@mjdennis thank you very much!

zyberg · Answer

@ILovePuppies the exclamation sign means factorial.