Find inverse of 15 modulo 88. Here the question: Find an inverse $a$ for $15$ modulo $88$ so that $0 \le a \le 87$; that is, find an integer $a \in \{0, 1, ..., 87\}$ so that $15a \equiv1$ (mod 88).
Here is my attempt to answer:
Find using the Euclidean Algorithm, we need to find $\gcd(88, 15)$, that must equal to $1$ to be possible to find an inverse of $15 \pmod{88}$.
\begin{align*}
88 & = 5 \times 15 + 13\\
15 & = 1 \times 13 + 2\\
13 & = 6 \times 2 + 1\\
2 & = 2 \times 1 + 0
\end{align*}
So,
$$\gcd(88, 15) = 1$$
Now, we need to write this into the form:
$$\gcd(88, 15) = 88x + 15y.$$
And find $x$ and $y$.
\begin{align*}
1 & = 13(1) + 2(-6)\\
& = 13(7) + 15(-6)\\
& = 88(7) + 15(-41)
\end{align*}
So, $x = 7$ and $y = -41$.
So, an inverse of $15 \pmod{88} = -41$. Now, I need to find an inverse that is between $0$ and $87$. What is a good easy approach to find other inverses? Any ideas please?

For a systemetic way, you have to use the Extended Euclidean Algorithm: if you have a Bézout's relation:
$$u\times 88+v\times 15=1$$
the you know the inverse of $15$ is $v\bmod88$.
The following shows how to display the computation, taking into account that, in Euclid's algorithm, the remainder at each step is a linear combination of $88$  and $15$:
\begin{array}[t]{lrrrr}
\text{Successive Divisions}& r_i & u_i & v_i & q_i\\
     \hline
 & 88 & 1 & 0 & \\
88 = {\color{red}5} \times 15 +\color{blue}{13}  & 15 &0 & 1 & \color{red}5 \\
15 = {\color{red}1} \times 13 + \color{blue}{2} & \color{blue}{13}  & 1 & -5 & \color{red}1 \\
13 = {\color{red}6} \times 2 + \color{blue}{1}  & \color{blue}{2} & -1 & 6 & \color{red}6 \\
 &  \color{blue}{1} & 7 & -41\\[2ex]
     \hline
    \end{array}
Thus the inverse of $15$ modulo $88$ is $\;-41\equiv 47\mod 88$.
