If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest
possible value of $p$.
1. We start with the given condition: when $2020$ is divided by a prime $p$, the remainder is $6$. This can be written in modular arithmetic as:
   \[
   2020 \equiv 6 \pmod{p}
   \]
   This implies:
   \[
   2020 - 6 \equiv 0 \pmod{p} \implies 2014 \equiv 0 \pmod{p}
   \]
   Therefore, $p$ must be a divisor of $2014$.

2. Next, we factorize $2014$ to find its prime factors:
   \[
   2014 = 2 \times 1007
   \]
   We then factorize $1007$:
   \[
   1007 = 19 \times 53
   \]
   Therefore, the prime factorization of $2014$ is:
   \[
   2014 = 2 \times 19 \times 53
   \]

3. The prime factors of $2014$ are $2$, $19$, and $53$. Among these, the largest prime factor is $53$.

Conclusion:
The largest possible value of $p$ is $\boxed{53}$.