The capacity of two pots are 240 litres and 112 litres respectively . Find the capacity of a container which can exactly measure the contents of the two pots .

Options

A. `9000 cm^(3)`
B. `12000 cm^(3)`
C. `16000 cm^(3)`
D. `8000 cm^(3)`

Answer

To find the capacity of a container that can exactly measure the contents of two pots with capacities of 240 liters and 112 liters, we need to determine the Least Common Multiple (LCM) of the two capacities. Here’s how to solve the problem step by step:

### Step 1: Find the prime factorization of both capacities.

**For 240 liters:**
- Divide by 2: \(240 \div 2 = 120\)
- Divide by 2: \(120 \div 2 = 60\)
- Divide by 2: \(60 \div 2 = 30\)
- Divide by 2: \(30 \div 2 = 15\)
- Divide by 3: \(15 \div 3 = 5\)
- Divide by 5: \(5 \div 5 = 1\)

So, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

**For 112 liters:**
- Divide by 2: \(112 \div 2 = 56\)
- Divide by 2: \(56 \div 2 = 28\)
- Divide by 2: \(28 \div 2 = 14\)
- Divide by 2: \(14 \div 2 = 7\)
- Divide by 7: \(7 \div 7 = 1\)

So, the prime factorization of 112 is:
\[ 112 = 2^4 \times 7^1 \]

### Step 2: Determine the LCM using the prime factorizations.

To find the LCM, we take the highest power of each prime number from the factorizations:
- For \(2\): The highest power is \(2^4\).
- For \(3\): The highest power is \(3^1\).
- For \(5\): The highest power is \(5^1\).
- For \(7\): The highest power is \(7^1\).

Thus, the LCM is:
\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 \]

### Step 3: Calculate the LCM.

Calculating step by step:
1. \(2^4 = 16\)
2. \(3^1 = 3\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now, multiply these together:
\[ 16 \times 3 = 48 \]
\[ 48 \times 5 = 240 \]
\[ 240 \times 7 = 1680 \]

Thus, the LCM of 240 and 112 is:
\[ \text{LCM} = 1680 \text{ liters} \]

### Step 4: Conclusion

The capacity of the container that can exactly measure the contents of both pots is **1680 liters**.

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