= Solution
Use the injective <group homomorphism> $F_2\hookrightarrow\operatorname{SL}_2(\mathbb Z)$ from part (a). If $1\ne g\in F_2$, its image is an integral matrix $M\ne I$. Choose a <prime number> $p$ that does not divide one nonzero entry of $M-I$. The <reduction modulo a prime in an integral matrix group> homomorphism
$$
\operatorname{SL}_2(\mathbb Z)\longrightarrow
\operatorname{SL}_2(\mathbb F_p)
$$
then sends $M$ to a nonidentity element. Its target is a <finite group>, so the composite map separates $g$ from the identity. Hence $F_2$ is a <residually finite group>.
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