Find the multiplicative inverses,if they exist for:(i)12 mod 15 (ii)12 mod 13; (if not, explain why they do not exist)
Can someone help me finish this? The first one doesn't have an inverse, because gcd(12, 15) is not 1, right? or Did I get it wrong?
Will appreciate pointing me to the right direction!
$12$ does not have an inverse modulo $15$ since $\gcd(12,15) \ne 1$.
Since $12 \equiv -1 \pmod{13}$, we have $12^2 \equiv (-1)^2 \equiv 1 \pmod{13}$ so $12$ is its own inverse modulo $13$. More generally, $n-1$ is always its own inverse modulo $n$ because $n-1 \equiv -1 \pmod{n}$.