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I am struggling to get this problem started. I have looked at similar problems in the book I am using for class (Discrete Mathematics with Applications, 7E) but none of them are seeming to help. Any help on how to solve these types of questions? I understand that I may need to use Fermat's Little Theorem but I am not seeing how to apply it completely.

charlieroth
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As you already mentioned, the first thing to do is use Fermat's Little Theorem, which says that for all primes $p$ and any $a$ that's relatively prime to $p$, $$ a^{p - 1} \equiv 1 \mod{p}$$ Notice in your example that $a = 21$ and $p = 47$ are indeed relatively prime, so $$21^{46} \equiv 1 \mod{47}$$ The next thing you want to do is notice that $4600 = 46(100)$. Recall that for any congruence $a \equiv b \mod{n}$, we also have that $a^k \equiv b^k \mod{n}$ for any $k \in \mathbb{N}$. In other words, we can "exponentiate" the congruence to the power of 100 to get $${21^{46}}^{100} \equiv 1^{100} \mod{47}$$ Or simply, $$21^{4600} \equiv 1 \mod{47}$$

AlkaKadri
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  • Now say I increase 4600 to 4601. Would I be able to apply the same method to this problem or would I need to approach it a different way? @AlkaKadri – charlieroth Nov 15 '17 at 23:06
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    We could start exactly the same way to end up with $21^{4600} \equiv 1 \mod{47}$. Recall that we can also "multiply" both sides of a congruence by any number. So afterwards, we'd simply have $21^{4600} \cdot 21 \equiv 1 \cdot 21 \mod{47} \Rightarrow 21^{4601} \equiv 21 \mod{47}$ – AlkaKadri Nov 15 '17 at 23:08
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    Notice that the approach is to get as close as we can to the desired exponent using some multiple of $p - 1$, and then then multiplying by an appropriate power of your base $a$ to take it the rest of the way. – AlkaKadri Nov 15 '17 at 23:09
  • Thanks! @AlkaKadri – charlieroth Nov 15 '17 at 23:12
  • You have a handy way of explaining things, @AlkaKadri. Thanks! – Defacto Nov 15 '17 at 23:18
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Hint: $${4600}=46\cdot100$$ and Fermat's Little Theorem.

DonAntonio
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I may need to use Fermat's Little Theorem

Indeed. It gives you $21^{46}\equiv 1\pmod{47}$, hence $21^{4600}\equiv 1^{100}=1\pmod{47}$ and you're done.

user502959
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