There are three heaps of rice weighing 120 kg, 144 kg and 204 kg. Find the maximum capacity of a bag so that the rice of each heap can be packed in exact number of bags.

Answer

To find the maximum capacity of a bag that can pack the rice from three heaps weighing 120 kg, 144 kg, and 204 kg in exact numbers, we need to determine the highest common factor (HCF) of these three weights. Here’s how to solve it step by step:

### Step 1: Prime Factorization
We will start by finding the prime factorization of each weight.

1. **For 120 kg:**
- Divide by 2: 120 ÷ 2 = 60
- Divide by 2: 60 ÷ 2 = 30
- Divide by 2: 30 ÷ 2 = 15
- Divide by 3: 15 ÷ 3 = 5
- Divide by 5: 5 ÷ 5 = 1
- So, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

2. **For 144 kg:**
- Divide by 2: 144 ÷ 2 = 72
- Divide by 2: 72 ÷ 2 = 36
- Divide by 2: 36 ÷ 2 = 18
- Divide by 2: 18 ÷ 2 = 9
- Divide by 3: 9 ÷ 3 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

3. **For 204 kg:**
- Divide by 2: 204 ÷ 2 = 102
- Divide by 2: 102 ÷ 2 = 51
- Divide by 3: 51 ÷ 3 = 17
- Divide by 17: 17 ÷ 17 = 1
- So, the prime factorization of 204 is:
\[ 204 = 2^2 \times 3^1 \times 17^1 \]

### Step 2: Identify Common Factors
Next, we identify the common prime factors from the factorizations:
- For \(2\): The minimum power is \(2^2\) (from 204).
- For \(3\): The minimum power is \(3^1\) (from both 120 and 204).

### Step 3: Calculate the HCF
Now, we multiply the common prime factors:
\[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Conclusion
The maximum capacity of a bag that can pack the rice from each heap in exact numbers is **12 kg**.

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