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Math Help - Show that 5|98^(150) + 3(103)^151

  1. #1
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    Question 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
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by steph3824 View Post
    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
    98=3 (mod\ 5)
    98^{150}=3^{150} (mod\ 5)
    3^2=-1 (mod\ 5)
    3^{150}=(-1)^{75}=-1 (mod\ 5)
    So 98^{150}=-1 (mod\ 5)

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

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