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 two tanks of 735 liters and 504 liters respectively, we need to determine the highest common factor (HCF) of the two quantities. Here’s a step-by-step solution:

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

**For 735:**
1. Divide by 3: \( 735 ÷ 3 = 245 \)
2. Divide by 5: \( 245 ÷ 5 = 49 \)
3. Divide by 7: \( 49 ÷ 7 = 7 \)
4. Divide by 7: \( 7 ÷ 7 = 1 \)

So, the prime factorization of 735 is:
\[ 735 = 3^1 \times 5^1 \times 7^2 \]

**For 504:**
1. Divide by 2: \( 504 ÷ 2 = 252 \)
2. Divide by 2: \( 252 ÷ 2 = 126 \)
3. Divide by 2: \( 126 ÷ 2 = 63 \)
4. Divide by 3: \( 63 ÷ 3 = 21 \)
5. Divide by 3: \( 21 ÷ 3 = 7 \)
6. Divide by 7: \( 7 ÷ 7 = 1 \)

So, the prime factorization of 504 is:
\[ 504 = 2^3 \times 3^2 \times 7^1 \]

### Step 2: Identify the common prime factors.

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

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

### Step 3: Calculate the HCF.

Now, we multiply 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 **21 liters**.

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