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!
(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?