# Thread: Show that 5|98^(150) + 3(103)^151

1. ## Show that 5|98^(150) + 3(103)^151

I would really really like some help with this please. I know that since 98=3 (mod 5) and 103=3 (mod 5) we have 98^150=3^150 (mod 5) and 103^151=3^151 (mod 5). I really don't have a clue where to go from here. I am having a pretty difficult time in my discrete math so please explain things clearly so that I will be able to understand what the heck is going on! Thanks so much for your help

2. Originally Posted by steph3824
I would really really like some help with this please. I know that since 98=3 (mod 5) and 103=3 (mod 5) we have 98^150=3^150 (mod 5) and 103^151=3^151 (mod 5). I really don't have a clue where to go from here. I am having a pretty difficult time in my discrete math so please explain things clearly so that I will be able to understand what the heck is going on! Thanks so much for your help
$\displaystyle 98=3 (mod\ 5)$
$\displaystyle 98^{150}=3^{150} (mod\ 5)$
$\displaystyle 3^2=-1 (mod\ 5)$
$\displaystyle 3^{150}=(-1)^{75}=-1 (mod\ 5)$
So $\displaystyle 98^{150}=-1 (mod\ 5)$

$\displaystyle 103=3 (mod\ 5)$
$\displaystyle 103^{151}=3^{151}(mod\ 5)$
$\displaystyle 3^{151}=-3 (mod\ 5)$
So $\displaystyle 103^{151}=-3 (mod 5)$

$\displaystyle 98^{150}+3(103)^{151}=-1+3*-3=-10=0 (mod\ 5)$
So $\displaystyle 5|98^{150}+3(103)^{151}$