To find the quotient of [tex]\( \frac{5}{31} \)[/tex] divided by [tex]\( \frac{15}{23} \)[/tex], follow these steps:
1. Rewrite the division as multiplication by the reciprocal:
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite:
[tex]\[
\frac{5}{31} \div \frac{15}{23} = \frac{5}{31} \times \frac{23}{15}
\][/tex]
2. Multiply the numerators and denominators:
Multiply the numerators together and the denominators together:
[tex]\[
\frac{5 \times 23}{31 \times 15} = \frac{115}{465}
\][/tex]
3. Simplify the fraction:
A fraction is simplified by dividing the numerator and denominator by their greatest common divisor (GCD). The GCD of 115 and 465 is:
[tex]\[
\gcd(115, 465) = 5
\][/tex]
Therefore, we divide both the numerator and the denominator by 5:
[tex]\[
\frac{115 \div 5}{465 \div 5} = \frac{23}{93}
\][/tex]
4. Confirm the simplified result:
The fraction [tex]\(\frac{23}{93}\)[/tex] is in its lowest terms because 23 is a prime number and does not divide 93 except by 1.
Therefore, the quotient of [tex]\( \frac{5}{31} \)[/tex] divided by [tex]\( \frac{15}{23} \)[/tex], reduced to the lowest fraction, is:
[tex]\[
\boxed{\frac{23}{93}}
\][/tex]
Answer: D.
