Une the ladder mathod to find the prime factors of these numbers. \( \begin{array}{ll}\text { a) } 180 & \text { b) } 78\end{array} \)

To find the prime factors of the numbers \(180\) and \(78\) using the ladder method, we will follow these steps: 1. **Start with the number**: Write the number at the top of the ladder. 2. **Divide by the smallest prime number**: Check if the number is divisible by the smallest prime (which is \(2\)). If it is, write the quotient below the number and continue dividing the quotient by the smallest prime until it is no longer divisible. 3. **Move to the next prime**: Once you can no longer divide by \(2\), move to the next smallest prime number and repeat the process. 4. **Continue until you reach \(1\)**: Keep dividing until you reach \(1\). The prime numbers used in the divisions are the prime factors of the original number. Let's apply this method to both numbers. ### a) Prime factors of \(180\) 1. Start with \(180\). 2. Divide by \(2\): - \(180 \div 2 = 90\) 3. Divide \(90\) by \(2\): - \(90 \div 2 = 45\) 4. \(45\) is not divisible by \(2\), so move to the next prime, which is \(3\): - \(45 \div 3 = 15\) 5. Divide \(15\) by \(3\): - \(15 \div 3 = 5\) 6. \(5\) is a prime number. So, the prime factors of \(180\) are \(2, 2, 3, 3, 5\) or in exponential form: \(2^2 \times 3^2 \times 5\). ### b) Prime factors of \(78\) 1. Start with \(78\). 2. Divide by \(2\): - \(78 \div 2 = 39\) 3. \(39\) is not divisible by \(2\), so move to the next prime, which is \(3\): - \(39 \div 3 = 13\) 4. \(13\) is a prime number. So, the prime factors of \(78\) are \(2, 3, 13\) or in exponential form: \(2 \times 3 \times 13\). ### Summary of Prime Factors - For \(180\): \(2^2 \times 3^2 \times 5\) - For \(78\): \(2 \times 3 \times 13\)