Inverse of $5$ under multiplication modulo $11$ on $Z_{11}$

Question

Find the inverse of $5$ under multiplication modulo $11$ on $Z_{11}$

Please help me with problem. Can't show any work because I don't know how to proceed.

You're looking for a number $n$ such that $5n \equiv 1 (\mod{11})$. — AJY, Jun 08 '16 at 18:01

score 2 · Answer 1 · answered Jun 08 '16 at 18:03

2

You have $11=5\times 2+1$, so $-2\times 5=1 \pmod{11}$.
So the inverse of $5$ in $Z_{11}$ is $-2=9 \pmod{11}$.

answered Jun 08 '16 at 18:03

Bérénice

Nice trick. Better than my "$11,22,33,44,$ aha! $45/5=9$" – Joffan Jun 08 '16 at 18:22

score 0 · Answer 2 · answered Jun 08 '16 at 18:18

0

if you know fermat's little theorem $a^{p-1} \equiv 1 \mod p $.

Then you can use this fact since $$a^{p-2} \cdot a = a \cdot a^{p-2} \equiv 1 \mod p$$

so just calculate $a^{p-2} \mod p$

in your case $5^9 \mod 11 = 9$

answered Jun 08 '16 at 18:18

user315567

2 Answers2