Question about computing $5^{64} \pmod{193}$

Question

How to compute $5^{64} \pmod{193}$?

My idea was:

$5^{64} = (5^4)^{16} \equiv 625^{16} \equiv 46^{16}\pmod{193}$.

Now I don't understand why it's $46^{16} \equiv (-7)^8 \equiv 49^4 \equiv 85^2 \equiv 84 \pmod{193}$.

How to get it here and how can the power be reduced?

You should understand why since it's your idea, and you get the final answer ;) — Adola, Jun 01 '20 at 16:01
If it was your idea then who told you that it was $46^{16}\equiv (-7)^8...etc$. — fleablood, Jun 02 '20 at 07:01

score 1 · Accepted Answer · answered Jun 02 '20 at 06:36

1

first we compute $5^8\,(mod193)$ az you computed: $$5^4\equiv46$$ therefore we have:$$5^8\equiv(-7)$$ $$5^{64}\equiv{(5^8)}^8\equiv{(-7)}^8\equiv7^8\equiv84\,(mod193)$$

answered Jun 02 '20 at 06:36

Mojbn

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Question about computing $5^{64} \pmod{193}$

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