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?