To solve this, we need to determine the integers from $1$ to $2024$ that can be represented in both forms $4a + 19$ and $6b + 21$, where $a$ and $b$ are integers. Let's go through the steps:

1. Identify the forms:

   • $x = 4a + 19$: This implies that $x \equiv 19 \pmod{4}$.
   • $x = 6b + 21$: This implies that $x \equiv 21 \pmod{6}$.

2. Rewrite the congruences:

   • Since $21 \equiv 3 \pmod{6}$, we can rewrite the second congruence as: $x \equiv 3 \pmod{6}$

   We now have the two conditions: $x \equiv 19 \pmod{4} \quad \text{and} \quad x \equiv 3 \pmod{6}.$

3. Use the Chinese Remainder Theorem (CRT): We need to find an integer $x$ that satisfies both conditions. This will provide a single congruence modulo the least common multiple of $4$ and $6$, which is $12$.

   • Rewrite the first condition as: $x \equiv 3 \pmod{6}$