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Find the order of 15 mod 257.

 May 9, 2019
 #1
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Find the order of 15 mod 257.

 

1. 257 is a prime and
2. \(257-1 = 2^8\) is a power of a prime, so
3. the order of 257 must also be a power of two. The order is \(2^k\) for some \(k= 0,1,2,\ldots, 8\)

 

Test:

\(\begin{array}{|r|r|l|} \hline k & 2^k \\ \hline 0 & 1 & 15^1 \equiv 15 \pmod{257} \\ 1 & 2 & 15^2 \equiv 225 \pmod{257} \\ 2 & 4 & 15^4 \equiv 253 \pmod{257} \\ 3 & 8 & 15^8 \equiv 16 \pmod{257} \\ 4 & 16 & 15^{16} \equiv 256 \pmod{257} \\ 5 & \color{red}32& 15^{32} \equiv {\color{red}1} \pmod{257} \\ \hline \end{array}\)

 

The order of 15 mod 257 is 32

 

source see: https://math.stackexchange.com/questions/2271153/calculate-the-order-of-3-modulo-257

 

laugh

 May 10, 2019

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