To multiply the fractions \( \frac{6}{7} \) and \( \frac{5}{6} \), you multiply the numerators together and the denominators together:
\[ \frac{6}{7} \times \frac{5}{6} = \frac{6 \times 5}{7 \times 6} = \frac{30}{42} \]
Next, we simplify \( \frac{30}{42} \). To do this, we need to find the greatest common divisor (GCD) of 30 and 42.
The factors of 30 are:
- 1, 2, 3, 5, 6, 10, 15, 30
The factors of 42 are:
- 1, 2, 3, 6, 7, 14, 21, 42
The GCD of 30 and 42 is 6. We can now divide the numerator and the denominator by their GCD:
\[ \frac{30 \div 6}{42 \div 6} = \frac{5}{7} \]
Thus, the answer in simplest form is
\[ \boxed{\frac{5}{7}} \]