Math Is Fun Forum

Chriskay

Guest

Number Theory

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

My teacher gave me the hint of n^5-n. which I know how to prove is divisible by 5.

But how do I reduce this problem down to n^5-n..Thanks!!!

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Number Theory

Well,you can say this:

a^5+b^5+c^5+d^5+d^5=
(a^5+b^5+c^5+d^5+d^5)-(a+b+c+d+e)=
(a^5-a)+(b^5-b)+(c^5-c)+(d^5-d)+(e^5-e) and this you can show is divisible by 5.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Chriskay

Guest

Re: Number Theory

Thanks man! I thought of doing this but couldn't think of a reason allowing it.  Obviously subtracting the string of integers is essentially just subtracting 0.  Thanks!

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Number Theory

You are welcome.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.