Modular Arithmetic Question (Rosen Discrete Mathematics and its applications)

Question

I came across this exercise in Rosen Discrete Mathematics and its applications and even after spending an hour plus googling I couldn't find an answer that could explain how this question is to be done. I saw somewhere that a certain theorem is used, but I'm not sure how it is applied.

$(19^3\mod23)^2\mod31$

The answer is: $((-4)^3 \mod 23)^2\mod31=(-64\mod23)^2\mod31=25$

Possibly useful theorem:

If $a \equiv b\mod m, c\equiv d\mod m$ then $ac=bd(\mod m)$

My question: I'm concerned with how I can get from $19^3$ to $(-4)^3$.

Thank you!

$19$ and $19 \pm 23k$ leave the same remainder when divided by $23$ — AgentS, Nov 01 '19 at 16:14
Also, $19^a$ and $(19\pm 23k)^a$ leave the same remainder when divided by $23$. — AgentS, Nov 01 '19 at 16:17
The composed mod operators look fishy. What is the exact statement of the problem? You may be misunderstanding relational vs operational mod. — Bill Dubuque, Nov 01 '19 at 17:58
@BillDubuque the exact statement is "33. Find each of these values." and exactly as what I have written above — IceTea, Nov 02 '19 at 16:42
Page 245, 4.2 Integer Representations and Algorithms, Qn 33c — IceTea, Nov 02 '19 at 16:46
Thank you for all your comments, both answers were helpful but I had to pick the one that directly answered my question. — IceTea, Nov 02 '19 at 16:49

score 1 · Accepted Answer · answered Nov 01 '19 at 16:13

1

It's because $19 \equiv -4 \pmod{23}$ (or equivalently $23|(19-(-4))$), hence $19^3 \equiv (-4)^3 \pmod{23}$

answered Nov 01 '19 at 16:13

Andronicus

3,436

Bill Dubuque · Answer 2 · 2019-11-01T18:15:16.170

1

You apply the given Congruence Product Rule. When we iterate it on the same congruence (i.e. by induction) we obtain the Congruence Power Rule $\ \bbox[5px,border:1px solid #c00]{a\equiv b\,\Rightarrow\, a^n\equiv b^n}\, $ for all $\,n\in\Bbb N.\,$ Therefore

$$\bmod 23 \!:\,\ 19\equiv -4\,\Rightarrow\, 19^3\equiv (-4)^3$$

edited Nov 01 '19 at 18:15

answered Nov 01 '19 at 17:51

Bill Dubuque

272,048

Modular Arithmetic Question (Rosen Discrete Mathematics and its applications)

2 Answers2