Thread: Prove 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!

Originally Posted by quah13579

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, 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 is divisible by 10... which means for some

Consider (we just rearranged here and cancelled the 1's)

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



Now show 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?

Originally Posted by quah13579

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!

is divisible by 5 by Fermat’s little theorem. It is also even (as and are either both even or both odd). Hence is divisible by 10.