La'Shondra starts by solving the inequality \( 6 + 3x \leq 24 \) by subtracting 6 from both sides. This gives her:
\[ 3x \leq 24 - 6 \]
Simplifying that, we have:
\[ 3x \leq 18 \]
Next, to isolate \( x \), she needs to divide both sides of the inequality by the coefficient of \( x \), which is 3. Therefore, the number she divides by is:
\[ 3 \]
So, the number is \( 3 \).