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The complex numbers $z_1$ and $z_2$ are given by
\begin{aligned} &z_1=5+3 \mathrm{i} \\ &z_2=1+p \mathrm{i} \end{aligned}
where $p$ is an integer.
(a) Find $\frac{\mathrm{z}_2}{\mathrm{z}_1}$ in the form $a+\mathrm{i} b$, where $a$ and $b$ are expressed in terms of $p$.

Given that $\arg \left(\frac{z_2}{z_1}\right)=\frac{\pi}{4}$,
(b) find the value of $p$.
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