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

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

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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: $$2^2 	imes 3^2 	imes 5$$.

### 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: $$2 	imes 3 	imes 13$$.

### Summary of Prime Factors
- For $$180$$: $$2^2 	imes 3^2 	imes 5$$
- For $$78$$: $$2 	imes 3 	imes 13$$
Prime factors of 180: 2, 2, 3, 3, 5 Prime factors of 78: 2, 3, 13

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

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