Here's a problem that i can't answer, pls help:
Find the remainder when 23^125 - 4*23^82 + 5*23^28 + 23^18 is divided by 23^2 + 23 +1.
You can use long division, although it is a painful process. I would show you but I have to learn the LaTex involved in long division.
Also, this is not a polynomial. There are no variables, so it is much easier than you think. Simply find the values of the two expressions, then use long division.
Is this correct? btw the answer is 0
$\displaystyle 23 / (23^2 + 24)$ = remainder: 23
$\displaystyle 23^2 / (23^2 + 24)$ = remainder: 529
$\displaystyle 23^3 / (23^2 + 24)$ = remainder: 1
$\displaystyle 23^4 / (23^2 + 24)$ = remainder: 23
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so $\displaystyle 23^{125} / (23^2 + 24)$ = remainder: 529
----125/3 = 41 r:2
$\displaystyle 4*23^{82} / (23^2 + 24)$ = remainder: 23
----82/3 = 27 r:1
$\displaystyle 5*23^{28} / (23^2 + 24)$ = Remainder: 23
----28/3 = 9 r:1
$\displaystyle 23^{18} / (23^2 + 24)$ = Remainder: 1
----18/3 = 6 r:0
then 529 - 4*23 + 5*23 + 1 = 553...which is divisible by
$\displaystyle 23^2 + 23 + 1 = 553$
therefore the remainder is 0 / no remainder