2. Use the ladder method to find the prime These numbers. \( \begin{array}{ll}\text { a) } 120 & \text { b) } 78 \\ \text { c) } 154 & \text { d) } 200 \\ \text { e) } 186 & \text { f) } 99\end{array} \)

The **ladder method** is a systematic way to determine the prime factors of a number by repeatedly dividing by the smallest possible prime number until you reach 1. Below are the prime factorizations for the given numbers using the ladder method: --- ### **a) 120** ``` 120 | 2 60 | 2 30 | 2 15 | 3 5 | 5 1 ``` **Prime Factors:** \(120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3 \times 5\) --- ### **b) 78** ``` 78 | 2 39 | 3 13 | 13 1 ``` **Prime Factors:** \(78 = 2 \times 3 \times 13\) --- ### **c) 154** ``` 154 | 2 77 | 7 11 | 11 1 ``` **Prime Factors:** \(154 = 2 \times 7 \times 11\) --- ### **d) 200** ``` 200 | 2 100 | 2 50 | 2 25 | 5 5 | 5 1 ``` **Prime Factors:** \(200 = 2 \times 2 \times 2 \times 5 \times 5 = 2^3 \times 5^2\) --- ### **e) 186** ``` 186 | 2 93 | 3 31 | 31 1 ``` **Prime Factors:** \(186 = 2 \times 3 \times 31\) --- ### **f) 99** ``` 99 | 3 33 | 3 11 | 11 1 ``` **Prime Factors:** \(99 = 3 \times 3 \times 11 = 3^2 \times 11\) --- **Summary of Prime Factors:** - **120:** \(2^3 \times 3 \times 5\) - **78:** \(2 \times 3 \times 13\) - **154:** \(2 \times 7 \times 11\) - **200:** \(2^3 \times 5^2\) - **186:** \(2 \times 3 \times 31\) - **99:** \(3^2 \times 11\) Each factorization breaks down the original number into its constituent prime numbers using the ladder method, ensuring that all factors are prime and multiplied together to reconstruct the original number.