+0

# Find the order of 15 mod 257.

-1
91
1
+79

Find the order of 15 mod 257.

May 9, 2019

#1
+22574
+2

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

May 10, 2019