To solve the inequality \( 6 + 3x \leq 24 \), La’Shondra starts by subtracting 6 from both sides:
\[ 6 + 3x - 6 \leq 24 - 6 \]
This simplifies to:
\[ 3x \leq 18 \]
Next, to isolate \( x \), she needs to divide both sides by 3.
So, the number she divides both sides by is:
\[ \text{The number is } 3. \]