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Say I have a problem like 2^191 divided by 5, how can I solve this using only modular arithmetic?

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    Tell us what you know about modular arithmetic and how you got started. Can you solve the problem for a few smaller exponents? Give it a try, report by editing your question. Then maybe we can help instead of downvoting. – 2017-02-14
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    It helps to note that $2^4\equiv 1 \pmod 5$. Can you take it from there? – 2017-02-14
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    and why 2^4≡1(mod5) do you convert to binary form to know that? – 2017-02-14
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    Just write it down! $2^4=16$ and $16=15+1$. – 2017-02-14
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    ok thanks but why 15+1, why not 14+2? – 2017-02-14
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    @Simon We want $\mod 5$, not $\mod 14$ or $\mod 7$. – 2017-02-14
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    I think you need to review the basic definitions. I'm sure whatever text you are using goes over these things. – 2017-02-14
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    so does 15 represent the time occurring of 5? wtch is 3 times? – 2017-02-14

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Here note that $2^4\equiv 1 \pmod 5 \implies (2^4)^{47} \equiv 1^{47} \pmod 5 \implies (2^4)^{47} \times 2^3 \equiv 1^{47} \times 8 \pmod 5 \\ \implies 2^{191} \equiv 3\pmod 5. $

$2^4 = 16 = 15+1$.

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    thanks but how did you get 2^4 to =1? – 2017-02-14
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    @Simon $16 \equiv 1 \pmod {5}$. – 2017-02-14
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    @Simon Have your learned yet about the *order* of an invertible element $a$ mod $n,\,$ i.e. the least natural number $k$ such that $\,a^{\large k}\equiv 1\pmod n\,?\ \ $ – 2017-02-14
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    @BillDubuque no, Im at the very begning at this – 2017-02-14