Two tanks contain 735 litres and 504 litres of water resepectively. Find the maximum capacity of a container which can measure the water of either tank an exact number of times.

Answer

To find the maximum capacity of a container that can measure the water from both tanks (735 liters and 504 liters) an exact number of times, we need to determine the highest common factor (HCF) of the two quantities. Here’s how we can do that step by step:

### Step 1: Factorize 735
To find the HCF, we first need to factorize 735 into its prime factors.

1. Divide 735 by 5 (the smallest prime number that divides it):
\[ 735 \div 5 = 147 \]
2. Next, factor 147:
\[ 147 \div 3 = 49 \]
3. Finally, factor 49:
\[ 49 = 7 \times 7 \]

So, the prime factorization of 735 is:
\[ 735 = 5 \times 3 \times 7^2 \]

### Step 2: Factorize 504
Now, we will factorize 504.

1. Divide 504 by 2:
\[ 504 \div 2 = 252 \]
2. Divide 252 by 2:
\[ 252 \div 2 = 126 \]
3. Divide 126 by 2:
\[ 126 \div 2 = 63 \]
4. Now, factor 63:
\[ 63 \div 3 = 21 \]
5. Finally, factor 21:
\[ 21 = 3 \times 7 \]

So, the prime factorization of 504 is:
\[ 504 = 2^3 \times 3^2 \times 7 \]

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from both factorizations:

- From 735: \(5^1, 3^1, 7^2\)
- From 504: \(2^3, 3^2, 7^1\)

The common prime factors are:
- For \(3\): The minimum power is \(3^1\)
- For \(7\): The minimum power is \(7^1\)

### Step 4: Calculate HCF
Now, we can calculate the HCF by multiplying the common prime factors:
\[ \text{HCF} = 3^1 \times 7^1 = 3 \times 7 = 21 \]

### Conclusion
The maximum capacity of the container that can measure the water from either tank an exact number of times is:
\[ \text{Maximum Capacity} = 21 \text{ liters} \]

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