Congruence solution $2^{14}= x (mod 17)

Question

How to solve the following congruence using properties.

$2^{14}=x (mod 17) $

I got the answer 13 using manual calculation. Thank in advance.

I would classify all the procedures to compute a remainder class of $2^{14}$ modulo $17$ as "manual calculations". — Sassatelli Giulio, Sep 22 '22 at 08:00

score 2 · Answer 1 · answered Sep 22 '22 at 07:57

2

Use fermat's little theorem, you only get $2^{16} \equiv 1 \space \mod 17$ It is not much use for this question.

So I just use $2^4 \equiv -1 \space \mod 17$ ,manual calculation like you.

$2^{14} \equiv (-1)^3 \times 2^2 \equiv -4 \equiv 13 \space \mod 17$

answered Sep 22 '22 at 07:57

Abel Wong

Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Sep 22 '22 at 11:11

1 Answers1