Three different containers contain 403 L, 434 L and 465 L of milk respectively. Find the capacity of the container which can measure the milk of all the containers in an exact number of times.

Answer

To find the capacity of the container that can measure the milk of all three containers (403 L, 434 L, and 465 L) in an exact number of times, we need to determine the highest common factor (HCF) of these three quantities. Here’s how to do it step by step:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 403:**
- 403 is divisible by 13 (403 ÷ 13 = 31).
- So, the prime factorization of 403 is \( 13 \times 31 \).

2. **Prime Factorization of 434:**
- 434 is divisible by 2 (434 ÷ 2 = 217).
- 217 is divisible by 7 (217 ÷ 7 = 31).
- So, the prime factorization of 434 is \( 2 \times 7 \times 31 \).

3. **Prime Factorization of 465:**
- 465 is divisible by 3 (465 ÷ 3 = 155).
- 155 is divisible by 5 (155 ÷ 5 = 31).
- So, the prime factorization of 465 is \( 3 \times 5 \times 31 \).

### Step 2: Identify Common Factors

Now, we look for the common prime factors in all three factorizations:

- **403:** \( 13 \times 31 \)
- **434:** \( 2 \times 7 \times 31 \)
- **465:** \( 3 \times 5 \times 31 \)

The only common factor among all three numbers is **31**.

### Step 3: Conclusion

Thus, the highest common factor (HCF) of 403, 434, and 465 is **31**. This means the capacity of the container that can measure the milk of all the containers in an exact number of times is **31 liters**.

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