To solve the inequality \( 6 + 3x \leq 24 \), La'Shondra starts by subtracting 6 from both sides. This gives:
\[ 3x \leq 24 - 6 \]
Calculating the right side:
\[ 3x \leq 18 \]
Next, to isolate \( x \), she will divide both sides of the inequality by 3.
The number she divides by is:
\[ 3 \]
So, the answer is \( \boxed{3} \).