#### Monoxdifly

The sum of ten integers is 0. Show that the sum of the fifth powers of these numbers is divisible by 5.

For this one I don't know what I have to do at all other than brute-forcing which may even be impossible.

#### romsek

MHF Helper
Where are you getting these problems? I'm beginning to suspect you are just throwing random stuff out to waste people's time.

#### Monoxdifly

I saved them to use later at my job (making students' worksheet) since they seemed relatively easy at a first glance. However, those questions turned to be difficult, so I need some assistance.

If you feel that this is wasting your time, why bother commenting in the first place? I'll just wait for the other people who don't think negatively to help.

#### Idea

$$\displaystyle x^5\equiv x \pmod 5$$

#### romsek

MHF Helper
I saved them to use later at my job (making students' worksheet) since they seemed relatively easy at a first glance. However, those questions turned to be difficult, so I need some assistance.

If you feel that this is wasting your time, why bother commenting in the first place? I'll just wait for the other people who don't think negatively to help.
well let's see

second question had an absurdly complicated answer that would take a human hours if not days to figure out

pardon me if I suspect your 3rd question

I will in fact ignore you from now on. Thank you.

#### Monoxdifly

$$\displaystyle x^5\equiv x \pmod 5$$
Ummm... Care to elaborate?

well let's see

second question had an absurdly complicated answer that would take a human hours if not days to figure out

pardon me if I suspect your 3rd question

I will in fact ignore you from now on. Thank you.
You know, you can ignore people without suspecting them wasting your time on purpose.

#### Idea

By Fermat's Theorem or by some other more elementary method we have

$$\displaystyle x^5\equiv x \pmod 5$$ meaning $$\displaystyle x^5- x$$ is divisible by $$\displaystyle 5$$ for all integers $$\displaystyle x$$

so that

$$\displaystyle \sum _i a_i=0$$

implies

$$\displaystyle \sum _i a_i^5\equiv \sum _i a_i=0$$

#### Monoxdifly

I think I kinda have a grasp now. Thanks.