Determine the HCF of 36 and 96 . (Use the ladder method)

To determine the Highest Common Factor (HCF) of 36 and 96 using the ladder method, we will follow these steps: 1. **Set up the ladder**: Write the two numbers side by side. 2. **Divide by the smallest prime number**: Start dividing both numbers by the smallest prime number that can divide at least one of them. 3. **Continue dividing**: If both numbers can be divided by the same prime number, divide both. If only one can be divided, divide that one. If neither can be divided by the current prime, move to the next prime number. 4. **Repeat until both numbers are reduced to 1**. 5. **Multiply the common prime factors**: The HCF is the product of all the prime factors that were used to divide both numbers. Let's perform the ladder method step by step: 1. **Set up the ladder**: ``` 36 | 96 ``` 2. **Divide by 2** (the smallest prime number): ``` 36 ÷ 2 = 18 96 ÷ 2 = 48 ``` So, we have: ``` 18 | 48 ``` 3. **Divide by 2 again**: ``` 18 ÷ 2 = 9 48 ÷ 2 = 24 ``` Now we have: ``` 9 | 24 ``` 4. **Next, divide by 3** (the next smallest prime number): ``` 9 ÷ 3 = 3 24 ÷ 3 = 8 ``` Now we have: ``` 3 | 8 ``` 5. **Finally, divide by 3 again**: ``` 3 ÷ 3 = 1 8 ÷ 2 = 4 ``` Now we have: ``` 1 | 4 ``` 6. **Divide by 2**: ``` 4 ÷ 2 = 2 ``` Now we have: ``` 1 | 2 ``` 7. **Divide by 2 again**: ``` 2 ÷ 2 = 1 ``` Now we have: ``` 1 | 1 ``` Now, we have completed the ladder method. The prime factors we used to divide both numbers are: - 2 (used twice) - 3 (used once) 8. **Multiply the common prime factors**: \[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \] Thus, the HCF of 36 and 96 is **12**.