Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Find the answer

1. Using the properties of indices and Example 1 in §3, construct a table of indices modulo 23 with base 11.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

1. From $\gamma_{23,11}(5) \cdot \gamma_{23,5}(11) \equiv 1(\bmod 22)$ , we get $\gamma_{23,11}(5)=5$ , hence $\gamma_{23,11}(a) \equiv 5 \gamma_{23,5}(a)(\bmod 22)$ .
$\text{a -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 23.11 (a) 12 4 17 19 18 18 16 9 3 21 11 a 1 2 3 4 5 6 7 8 9 10 11 23.11 (a) 0 10 14 20 5 2 7 8 6 15 1}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.