Three different containers contains different qualities of mixtures of milk and water, whose measurement are 403 kg, 434 kg and 464 kg .What biggest measure must be there to measure all the different quantities exactly

Options

A. 1 kg
B. 7 kg
C. 31 kg
D. 41 kg

Answer

To find the biggest measure that can be used to measure all the different quantities of the mixtures in the containers, we need to determine the Highest Common Factor (HCF) of the three quantities: 403 kg, 434 kg, and 464 kg.

### Step-by-Step Solution:

1. **Find the Prime Factorization of Each Quantity:**
- **403 kg:**
- Check divisibility by 2: 403 is odd, so not divisible by 2.
- Check divisibility by 3: The sum of the digits (4 + 0 + 3 = 7) is not divisible by 3.
- Check divisibility by 5: 403 does not end in 0 or 5.
- Check divisibility by 7: \( 403 ÷ 7 = 57.57 \) (not divisible).
- Check divisibility by 11: \( 403 ÷ 11 = 36.64 \) (not divisible).
- Check divisibility by 13: \( 403 ÷ 13 = 31 \) (403 = 13 × 31).
- Thus, the prime factorization of 403 is \( 13 × 31 \).

- **434 kg:**
- Check divisibility by 2: \( 434 ÷ 2 = 217 \) (434 = 2 × 217).
- Check divisibility of 217:
- Not divisible by 2, 3, or 5.
- Check 7: \( 217 ÷ 7 = 31 \) (217 = 7 × 31).
- Thus, the prime factorization of 434 is \( 2 × 7 × 31 \).

- **464 kg:**
- Check divisibility by 2: \( 464 ÷ 2 = 232 \) (464 = 2 × 232).
- Check divisibility of 232: \( 232 ÷ 2 = 116 \) (232 = 2 × 116).
- Check divisibility of 116: \( 116 ÷ 2 = 58 \) (116 = 2 × 58).
- Check divisibility of 58: \( 58 ÷ 2 = 29 \) (58 = 2 × 29).
- Thus, the prime factorization of 464 is \( 2^4 × 29 \).

2. **List the Prime Factors:**
- 403 = \( 13 × 31 \)
- 434 = \( 2 × 7 × 31 \)
- 464 = \( 2^4 × 29 \)

3. **Identify Common Prime Factors:**
- The only common prime factor among 403, 434, and 464 is 31.

4. **Determine the HCF:**
- Since 31 is the only common prime factor, the HCF of 403, 434, and 464 is 31.

### Conclusion:
The biggest measure that must be there to measure all the different quantities exactly is **31 kg**.