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The question asks to find the **HCF (Highest Common Factor)** and **LCM (Lowest Common Multiple)** of the given sets of numbers using **short division**. Let's solve each one:

### 1. Find HCF and LCM of 36, 81, and 180:
**Short division** method works by dividing the numbers by common prime factors until we can't divide further. 

**Step-by-step:**
- Divide by 3:  
   36 ÷ 3 = 12,  
   81 ÷ 3 = 27,  
   180 ÷ 3 = 60  
   (Numbers left: 12, 27, 60)

- Divide by 3 again:  
   12 ÷ 3 = 4,  
   27 ÷ 3 = 9,  
   60 ÷ 3 = 20  
   (Numbers left: 4, 9, 20)

- Now, we cannot divide by 3 anymore. Try dividing by the next prime number, 2:

- Divide by 2:  
   4 ÷ 2 = 2,  
   (9 and 20 are not divisible by 2)  
   (Numbers left: 2, 9, 20)

- Divide by 2 again:  
   2 ÷ 2 = 1,  
   (9 and 20 remain as they are)  
   (Numbers left: 1, 9, 20)

- Divide by 5 (a common divisor for 20):  
   9 remains the same,  
   20 ÷ 5 = 4  
   (Numbers left: 1, 9, 4)

- Continue until no further common divisors are available.

From the above steps:
- The **HCF** is the product of the divisors used: \( HCF = 3 \times 3 = 9 \).
- The **LCM** is the product of the HCF and all remaining numbers:  
  \( LCM = 9 \times 12 \times 60 = 540 \).

### 2. Find HCF and LCM of 48, 72, and 128:
Using the same method:

- Divide by 2:  
  48 ÷ 2 = 24,  
  72 ÷ 2 = 36,  
  128 ÷ 2 = 64  
  (Numbers left: 24, 36, 64)

- Divide by 2 again:  
  24 ÷ 2 = 12,  
  36 ÷ 2 = 18,  
  64 ÷ 2 = 32  
  (Numbers left: 12, 18, 32)

- Continue this division process.

Final results:
- **HCF** = 8  
- **LCM** = 576

### 3. Find HCF and LCM of 90, 108, and 135:
Similar steps give us:
- **HCF** = 9
- **LCM** = 540

### 4. Find HCF and LCM of 36, 72, 90, and 108:
The steps are:
- **HCF** = 18  
- **LCM** = 2160

Would you like more detailed steps for any of these problems, or clarifications on the short division method?

Here are 5 related questions to think about:
1. How can prime factorization be used instead of short division to find HCF and LCM?
2. What are other methods for finding HCF and LCM?
3. Can you find the HCF of 54, 81, and 162?
4. How do we handle numbers that are co-prime?
5. Why is the product of HCF and LCM equal to the product of the two numbers for pairs?

**Tip:** The HCF is always a divisor of the LCM!

Find the HCF and LCM by using short division for the following sets of numbers: 1. 36, 81, 180 2. 48, 72, 128 3. 90, 108, 135 4. 36, 72, 90, 108

This problem helps students find the HCF and LCM of given sets of numbers using the short division method. Learn step-by-step how to solve for 36, 81, 180; 48, 72, 128; 90, 108, 135; and 36, 72, 90, 108. Ideal for Grades 6-8.

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