Prove n^5 - n is divisible by 240 when n is a positive odd integer, but ...

[lookagain]

not by induction. I did not post this in the original thread, because I did not want to hijack it.
I don't think the original problem as asked by the OP there should ask to do it by induction,
because I find the solution to be unwieldy.
(I thank Harry_the_cat for his multiple posts of explaining and working out many steps there.)

$\displaystyle n^5 - n \ = \$

$\displaystyle n(n^4 - 1) \ = \$

$\displaystyle n(n^2 - 1)(n^2 + 1) \ = \$

$\displaystyle n(n - 1)(n + 1)(n^2 + 1) \ = \$

$\displaystyle (n - 1)(n)(n + 1)(n^2 + 1)$

The expression is divisible by 3, because it is the product of three consecutive integers.

If the odd number ends in a 3 or 7, then its square will end in a 9. When added to 1 it
will be divisible by 5. So, the $\displaystyle \ (n^2 + 1) \$ factor will be divisible by 5.

If the odd number ends in a 1, then (n - 1) will be divisible by 5, if the odd number ends
in a 5, then n will be divisible by 5, or if the odd number ends in a 9, then (n + 1) will
be divisible by 5. So, in all cases of odd numbers, the expression is divisible by 5.

Let $\displaystyle \ n = 2m + 1, \ \$where m is an integer. Substitute that into the expression:

$\displaystyle [(2m + 1) \ - \ 1](2m + 1)[(2m + 1) \ + \ 1][(2m + 1)^2 \ + \ 1] \ = \$

$\displaystyle (2m)(2m + 1)(2m + 2)(4m^2 + 4m + 2) \ = \$

$\displaystyle (2)(2)(2)(m)(2m + 1)(m + 1)(2m^2 + 2m + 1) \ = \$

$\displaystyle 8(m)(m + 1)(2m + 1)(2m^2 + 2m + 1)$

The last expression is divisible by 8. Also, the last expression is divisible by an additional
2, because the product (m)(m + 1) is divisible by 2 as it is the product of two consecutive
integers. So, the expression is divisible by 16.

Taken together, the expression is divisible by 3, 5, and 16.

3*5*16 = 240

Therefore, $\displaystyle \ n^5 - n \$ is divisible by 240 when n is a positive odd integer.

 

[Harry_the_cat]

(I thank Harry_the_cat for his multiple posts of explaining and working out many steps there.)

"her" - just sayin'. But thanks for the thanks! ?