A fraction $$ \frac{a}{b} $$ can be written in simplest terms (that is, as $$ \frac{c}{d} $$, where $$ c $$ and $$ d $$ have no common factor other than 1$$ ) $$ by dividing both a and $$ b $$ by $$ G C D(a, b) $$. (a) Compute $$ G C D(30,228) $$. (b) Determine a fraction in simplest terms equivalent to $$ \frac{30}{228} $$. (a) $$ G C D(30,228)=\square $$

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1. We compute $$ GCD(30,228) $$ using the Euclidean algorithm.  
   - First, divide $$ 228 $$ by $$ 30 $$:  
     $$
     228 = 30 \times 7 + 18
     $$  
     So, $$ GCD(30,228) = GCD(30,18) $$.

2. Next, divide $$ 30 $$ by $$ 18 $$:  
   $$
   30 = 18 \times 1 + 12
   $$  
   So, $$ GCD(30,18) = GCD(18,12) $$.

3. Then, divide $$ 18 $$ by $$ 12 $$:  
   $$
   18 = 12 \times 1 + 6
   $$  
   So, $$ GCD(18,12) = GCD(12,6) $$.

4. Finally, divide $$ 12 $$ by $$ 6 $$:  
   $$
   12 = 6 \times 2 + 0
   $$  
   Since the remainder is $$ 0 $$, we have $$ GCD(12,6)=6 $$.

Thus,  
$$
GCD(30,228)=6.
$$

5. To express the fraction $$ \frac{30}{228} $$ in simplest terms, divide both the numerator and the denominator by $$ 6 $$:
   $$
   \frac{30 \div 6}{228 \div 6} = \frac{5}{38}.
   $$

(a) $$ GCD(30,228)=\boxed{6} $$.
$$ GCD(30,228)=6 $$. The fraction $$ \frac{30}{228} $$ simplifies to $$ \frac{5}{38} $$.

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