Solve $660x\equiv 1$ mod $43$

Question

Solve $660x\equiv 1$ mod $43$

I found $660\equiv 15$ mod $43$

So I want to solve $15x\equiv 1$ mod $43$

I tried some low numbers but none of them worked. I'm not sure what to do now.

you could try the extended Euclidean algorithm to get a Bezout relation — J. W. Tanner, Mar 18 '20 at 01:31
See my post in https://math.stackexchange.com/questions/407478/solving-a-linear-congruence/407482 — lab bhattacharjee, Mar 18 '20 at 01:42
$\bmod 43!:,\ \dfrac{1}{15}\equiv \dfrac{3}{45}\equiv \dfrac{46}{2}\equiv 23\ $ by Gauss's algorithm and modular halving. — Bill Dubuque, Mar 18 '20 at 01:55

score 1 · Accepted Answer · answered Mar 18 '20 at 01:24

1

Here's one way:

$15\times3=45\equiv2\bmod 43$

$2\times 22=44\equiv 1\bmod 43$

Therefore

$15\times 3\times22\equiv 1 \bmod 43$

$15\times 66\equiv1\bmod 43$

$15\times 23\equiv 1 \bmod 43$

answered Mar 18 '20 at 01:24

J. W. Tanner

This is essentially Gauss's algorithm (in non-fractional form). Follow the link in case it is not clear from above how it works generally (for prime moduli only). – Bill Dubuque Mar 18 '20 at 01:46

1 Answers1