SOLUTION: How do u prove this?...Show that 3 divides n^3-n for all positive integers n. And does 4 divide n^4-n for all positive intgers n? AND i need to use proof by induction i think, i g

Question 12319: How do u prove this?...Show that 3 divides n^3-n for all positive integers n. And does 4 divide n^4-n for all positive intgers n? AND i need to use proof by induction i think, i get the first part but then get stuck at the end...please help
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Show that 3 divides n^3-n for all positive integers n.
Proof: Claim: 3 isa divisor of for all integer
.....(**)
Basic: When n = 1 ,
So, (**) is true when
Induction Hypothesis: When (**) is true.
Hence, there exists an integer q such that .
Consider
=
=
=
This shows 3 is a divisor of .
And so,the inductive proof is complete.

[Actually, is a product of three consecutive positive integers
for any positive integer n. Why ? ]
And does 4 divide n^4-n for all positive intgers n?
When n =2,
Hence 4 cannot divide n^4-n in general.
In fact,
Clearly, 2 is a divisor of
We only can claim 2 divides for all positive n.
Kenny

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