Hi,

Could someone please explain how to do this:

Prove that n^5 - n is divisible by 10 for all positive integer n.

[Hint: a number is divisible by 10 if and only if it is divisible by 5 and ....]

thanks!

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- Aug 21st 2009, 07:05 PMquah13579Prove problem
Hi,

Could someone please explain how to do this:

Prove that n^5 - n is divisible by 10 for all positive integer n.

[Hint: a number is divisible by 10 if and only if it is divisible by 5 and ....]

thanks! - Aug 21st 2009, 07:35 PMartvandalay11
(your hint is incorrect with the double implication, while it is true that if a number is divisible by 10 then it is divisble by 5, it is not true to say that if a number is divisible by 5 then it is divisible by 10 for ex:25, though I'm guessing a 2 shoulda been after that "and")

Proof by induction

Base case n=1, $\displaystyle 1^5-1=1-1=0$ and 0 is divisible by 10 since 0=10*0

(an integer a is divisible by an integer b if there exists some c such that a=bc)

Now assume $\displaystyle n^5-n$ is divisible by 10... which means $\displaystyle n^5-n=10k$ for some $\displaystyle k\in\mathbb{Z}$

Consider $\displaystyle (n+1)^5-(n+1)=n^5+5n^4+10n^3+10n^2+5n+1-n-1=n^5-n+5n^4+10n^3+10n^2+5n$ (we just rearranged here and cancelled the 1's)

By our assumption we can substitute in for the first part of the expression so

$\displaystyle =10k+5n^4+10n^3+10n^2+5n=10(k+n^3+n^2)+5(n^4+n)$

Now show $\displaystyle n^4+n$ is divisible by 2 for all positive integers, then the 5*2 will give us a 10 and the entire expression will be divisible by 10

Can you do this induction? - Aug 22nd 2009, 03:17 AMTheAbstractionist