\begin{tabular}{|l|l|}\hline Determine the sum of the prime factors of 675. \\ \( \begin{array}{l}\text { A molecule is measured and the width is } 3 \times 10^{-6} \mathrm{~cm} \text {. How long will a string } \\ \text { of } 225 \text { similar molecules be? Give your answer in scientific notation. }\end{array} \) \\ \( \begin{array}{l}\text { A farmer has } 2400 \text { sheep. He loses } 32 \frac{1}{2} \% \text { due to the drought. After that he } \\ \text { loses } 15 \% \text { due to sickness. How many sheep does he have now? }\end{array} \) \\ \hline\end{tabular}

We’ll solve each problem step by step. ───────────────────────────── Problem 1: Sum of the Prime Factors of 675 Step 1. Factor 675.   • Divide 675 by 5: 675 ÷ 5 = 135   • Divide 135 by 5: 135 ÷ 5 = 27   • Recognize that 27 = 3 × 3 × 3 = 3³ So, the prime factorization of 675 is 3³ × 5². Step 2. Sum the distinct prime factors.   • The distinct prime factors are 3 and 5.   • Their sum is 3 + 5 = 8. ───────────────────────────── Problem 2: Length of a String of 225 Molecules Each molecule has a width of 3 × 10⁻⁶ cm. Step 1. Multiply the width of one molecule by 225:   • Total length = 225 × (3 × 10⁻⁶ cm) = 675 × 10⁻⁶ cm Step 2. Convert 675 × 10⁻⁶ cm to proper scientific notation.   • Write 675 as 6.75 × 10² (since 675 = 6.75 × 100)   • Now, total length = 6.75 × 10² × 10⁻⁶ cm = 6.75 × 10⁻⁴ cm ───────────────────────────── Problem 3: Number of Sheep Remaining A farmer starts with 2400 sheep, losing 32½% due to drought, then loses an additional 15% due to sickness. Step 1. Calculate the remaining sheep after the drought:   • Percentage remaining = 100% − 32.5% = 67.5%   • Sheep remaining = 67.5% of 2400 = 0.675 × 2400 = 1620 sheep Step 2. Calculate the remaining sheep after sickness:   • He loses 15% due to sickness, so 85% remain.   • Final number of sheep = 85% of 1620 = 0.85 × 1620 = 1377 sheep ───────────────────────────── Summary of Answers:   1. The sum of the prime factors of 675 is 8.   2. The total length of 225 molecules is 6.75 × 10⁻⁴ cm.   3. The farmer has 1377 sheep remaining.