Show that for all integer values of $\displaystyle x$,
. . $\displaystyle x^9 - 6x^7 + 9x^5 - 4x^3$ is divisible by $\displaystyle 8640.$
Soroban, am I on the right track?.
We have to show that $\displaystyle x^{9}-6x^{7}+9x^{5}-4x^{4}$ is divisible by $\displaystyle 8640=2^{6}\cdot{3^{3}}\cdot{5}$
Factor: $\displaystyle x^{3}(x-2)(x-1)^{2}(x+1)^{2}(x+2)$
I would say because 5 is a factor of $\displaystyle x(x-1)(x-2)(x+2)$
3 is a factor of $\displaystyle x(x-1)(x-2), \;\ x(x-1)(x+2), \;\ x(x+1)(x+2)$
8 is a factor of $\displaystyle x(x-2)(x-1)(x+1), \;\ x(x-1)(x+1)(x+2)$
Hello, Galactus!
Of course, you meant: 5 is a factor of $\displaystyle x(x-1)(x+1)(x-2)(x+2)$
Then your solution is correct.
The "Quickie" solution goes like this:
We know that the product of $\displaystyle n$ consecutive integers is divisible by $\displaystyle n$
. . and the product of four consecutive integers is divisible by $\displaystyle 2^3.$
We have: .$\displaystyle 8640 \:=\:2^6\!\cdot\!3^3\!\cdot\!5$
. . and: .$\displaystyle N \;=\;(x-2)\ (x-1)^2\ x^3\ (x+1)^2\ (x+2)$
$\displaystyle \text{Since }N \;=\;\underbrace{[(x-2)(x-1)x]}_{\text{3 consecutive}}\ \underbrace{[(x-1)x(x+1)]}_{\text{3 consecutive}}\ \underbrace{[x(x+1)(x+2)]}_{\text{3 consecutive}} $
. . then $\displaystyle N$ is divisible by $\displaystyle 3^3.$
$\displaystyle \text{Since }N \;=\;\underbrace{[(x-2)(x-1)x(x+1)(x+2)]}_{\text{5 consecutive}}\ [(x-1)x^2(x+1)]$
. . then $\displaystyle N$ is divisible by $\displaystyle 5.$
$\displaystyle \text{Since }N \;=\;x\ \underbrace{[(x-2)(x-1)x(x+1)]}_{\text{div by }2^3}\ \underbrace{[(x-1)x(x+1)(x+2)]}_{\text{div by }2^3}$
. . then $\displaystyle N$ is divisible by $\displaystyle 2^6.$
Therefore, $\displaystyle N$ is divisible by $\displaystyle 2^6\!\cdot\!3^3\!\cdot\!5\;=\;8640.$