The fifth power

Number Theory Level pending

Let $P(n)=n^5-n$ be a function defined for positive integers $n$ . Which of the following statements are true for all $n$ .

[1] $P(n) \mod (n+1) = 0$

[2] $P(n) \mod 10 = 0$

[3] $P(n) \mod (n^2+1) = 0$

[1] and [2], not [3] [1] and [3], not [2] [1] only [1], [2] and [3] None of the other options

1 solution

Janardhanan Sivaramakrishnan
Jan 26, 2015

First of all, $n^5-n=n(n-1)(n+1)(n^2+1)$ .

Hence, statements [1] and [3] are true.

Further, it can also be verified that $n^5 \mod 10 = n \mod 10$ for all positive integers $n$ . Thus, statement [2] is also true.