Use Euclid's algorithm to find the multiplicative inverse $11$ modulo $59$ I was wondering if this answer would be correct
the multiplicative of $11$ modulo $59$ would be $5$ hence $5\cdot11 \equiv 4 \pmod{59}$.
Is this correct?

No. A multiplicative inverse $a$ has $5 \cdot a \equiv 1 \pmod{59}$ and $11$ has $5 \cdot 11 \equiv -4 \not\equiv 1\pmod{59}$. If you want to find the inverse, use the Euclidean algorithm, as you said:
\begin{align*}
  59 &= 5 \cdot 11 + 4\\
  11 &= 2 \cdot 4  + 3\\
  4  &= 1 \cdot 3  + 1\\
\end{align*}
Hence
\begin{align*}
   1 &= -1 \cdot 3 + 4\\
     &= -1 \cdot (11 - 2 \cdot 4) + 4\\
     &= -1 \cdot 11 + 3 \cdot 4\\
     &= -1 \cdot 11 + 3 \cdot (59 - 5 \cdot 11)\\
     &= -16 \cdot 11 + 3 \cdot 59
\end{align*}
Hence, $\pmod{59}$, we have
$$ 1 \equiv -16 \cdot 11 \equiv 43 \cdot 11 \pmod{59} $$
So, the multiplicative inverse is $43$.
