To find the quotient of \( \frac{3}{7} \div 9 \), we can rewrite the division as multiplication by the reciprocal:
\[ \frac{3}{7} \div 9 = \frac{3}{7} \times \frac{1}{9} \]
Now, we multiply the fractions:
\[ \frac{3 \times 1}{7 \times 9} = \frac{3}{63} \]
Next, we simplify \( \frac{3}{63} \). We can do this by finding the greatest common divisor (GCD) of 3 and 63, which is 3.
Now, we divide the numerator and the denominator by their GCD:
\[ \frac{3 \div 3}{63 \div 3} = \frac{1}{21} \]
Thus, the simplified quotient of \( \frac{3}{7} \div 9 \) is
\[ \boxed{\frac{1}{21}}. \]
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