We start by finding the greatest common factor of $5$ and $10\mathbin{:}$

• Factors of $5\mathbin{:}\quad 1,{\color{blue}{5}}$

• Factors of $10\mathbin{:}\quad 1, 2, {\color{blue}{5}}, 10$

So, the greatest common factor of $5$ and $10$ is ${\color{blue}{5}}.$

Dividing both the numerator and denominator by ${\color{blue}{5}}$ gives the following:

\[ \dfrac{5}{10}=\dfrac{5\div {\color{blue}{5}}}{10 \div {\color{blue}{5}}}=\dfrac{1}{2} \]

Therefore, $\dfrac{5}{10}$ expressed in lowest terms is $\dfrac12.$

We start by finding the greatest common factor of $30$ and $9\mathbin{:}$

• Factors of $30\mathbin{:}\quad 1,2,{\color{blue}{3}},5,6,10,15,30$

• Factors of $9\mathbin{:}\quad 1,{\color{blue}{3}}, 9$

So, the greatest common factor of $30$ and $9$ is ${\color{blue}{3}}.$

Dividing both the numerator and denominator by ${\color{blue}{3}}$ gives the following:

\[ \dfrac{30}{9} = \dfrac{30 \div {\color{blue}{3}}}{9 \div {\color{blue}{3}}}=\dfrac{10}{3} \]

Therefore, $\dfrac{30}{9}$ expressed in lowest terms is $\dfrac{10}{3}.$

We start by finding the greatest common factor of $12$ and $26\mathbin{:}$

• Factors of $12\mathbin{:}\quad 1,{\color{blue}{2}}, 3, 4, 6, 12$

• Factors of $26\mathbin{:}\quad 1,{\color{blue}{2}}, 13, 26$

So, the greatest common factor of $12$ and $26$ is ${\color{blue}{2}}.$

Dividing both the numerator and denominator by ${\color{blue}{2}}$ gives the following:

\[ \dfrac{12}{26}=\dfrac{12\div {\color{blue}{2}}}{26 \div {\color{blue}{2}}}=\dfrac{6}{13} \]

Therefore, $\dfrac{12}{26}$ expressed in lowest terms is $\dfrac{6}{13}.$

We start by finding the greatest common factor of $18$ and $12{:}$

• Factors of $18{:}\quad 1, 2, 3, {\color{blue}{6}}, 9, 18$

• Factors of $12{:}\quad 1, 2, 3, 4, {\color{blue}{6}}, 12$

So, the greatest common factor of $18$ and $12$ is ${\color{blue}{6}}.$

Dividing both the numerator and denominator by ${\color{blue}{6}}$ gives the following: \[ \dfrac{18}{12} = \dfrac{18 \div {\color{blue}{6}}}{12 \div {\color{blue}{6}}} = \dfrac{3}{2} \]

Therefore, $\dfrac{18}{12}$ expressed in lowest terms is $\dfrac{3}{2}.$

A fraction is written in lowest terms if the greatest common factor of the numerator and denominator equals $1.$

With that in mind, let's examine our fractions.

• The fraction $\dfrac{4}{9}$ is written in lowest terms since the greatest common factor of $4$ and $9$ is $1.$

• The fraction $\dfrac{3}{9}$ in not written in lowest terms. Notice that the greatest common factor of $3$ and $9$ is $3.$ So, we can divide both the numerator and denominator by $3.$ \[ \dfrac{3}{9} = \dfrac{3\div 3}{9\div 3} = \dfrac13 \]

• The fraction $\dfrac{15}{4}$ is written in lowest terms since the greatest common factor of $15$ and $4$ is $1.$

Therefore, the correct answer is "I and III only."

A fraction is written in lowest terms if the greatest common factor of the numerator and denominator equals $1.$

With that in mind, let's examine our fractions.

• The fraction $\dfrac{44}{12}$ in not written in lowest terms. Notice that the greatest common factor of $44$ and $12$ is $4.$ So, we can divide both the numerator and denominator by $4.$ \[ \dfrac{44}{12} = \dfrac{44\div 4}{12\div 4} = \dfrac{11}{3} \]

• The fraction $\dfrac{17}{13}$ is written in lowest terms since the greatest common factor of $17$ and $13$ is $1.$

• The fraction $\dfrac{27}{36}$ is not written in lowest terms. Notice that the greatest common factor of $27$ and $36$ is $9.$ So, we can divide both the numerator and denominator by $9.$ \[ \dfrac{27\div 9}{36\div 9} = \dfrac{3}{4} \]

Therefore, the correct answer is "II only."