To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we can use the property that states:
\[ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \]
In this case, we let \( a = 125 \), \( b = 75 \), and \( m = 3 \). Therefore, we can rewrite the expression as:
\[ 125^3 \div 75^3 = \left(\frac{125}{75}\right)^3 \]
Next, we simplify the fraction \( \frac{125}{75} \). Both numbers can be divided by 25:
\[ \frac{125}{75} = \frac{125 \div 25}{75 \div 25} = \frac{5}{3} \]
Now, substituting back into our expression, we have:
\[ \left(\frac{125}{75}\right)^3 = \left(\frac{5}{3}\right)^3 \]
Now we calculate \( \left(\frac{5}{3}\right)^3 \):
\[ \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27} \]
Thus, the simplified expression is:
\[ \boxed{\frac{125}{27}} \]