Three different containers contain different quantities of mixtures of milk and wate r, whose measurements are 403 L, 434 L and 465 L. What biggest measure must be there to measure all the different quantities exactly?

Options

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

Answer

To find the biggest measure that can be used to measure the quantities of 403 L, 434 L, and 465 L exactly, we need to determine the greatest common divisor (GCD) or highest common factor (HCF) of these three numbers. Here’s how we can do that step by step:

### Step 1: Prime Factorization of Each Number
1. **Prime Factorization of 403**:
- 403 is not divisible by 2 (since it's odd).
- Check divisibility by 3: \(4 + 0 + 3 = 7\) (not divisible by 3).
- Check divisibility by 5 (last digit is not 0 or 5).
- Check divisibility by 7: \(403 ÷ 7 \approx 57.57\) (not divisible).
- Check divisibility by 11: \(403 ÷ 11 \approx 36.64\) (not divisible).
- Check divisibility by 13: \(403 ÷ 13 \approx 31\) (not divisible).
- Check divisibility by 17: \(403 ÷ 17 \approx 23.59\) (not divisible).
- Check divisibility by 19: \(403 ÷ 19 \approx 21.21\) (not divisible).
- Finally, check divisibility by 31: \(403 ÷ 31 = 13\) (exact).
- Thus, \(403 = 31 \times 13\).

2. **Prime Factorization of 434**:
- 434 is even, so divide by 2: \(434 ÷ 2 = 217\).
- Now factor 217: Check divisibility by 3 (not divisible).
- Check divisibility by 7: \(217 ÷ 7 \approx 31\) (exact).
- Thus, \(434 = 2 \times 7 \times 31\).

3. **Prime Factorization of 465**:
- 465 is odd, check divisibility by 3: \(465 ÷ 3 = 155\).
- Now factor 155: Check divisibility by 5: \(155 ÷ 5 = 31\).
- Thus, \(465 = 3 \times 5 \times 31\).

### Step 2: List the Prime Factors
- **403**: \(31 \times 13\)
- **434**: \(2 \times 7 \times 31\)
- **465**: \(3 \times 5 \times 31\)

### Step 3: Identify Common Factors
The common prime factor among all three numbers is **31**.

### Step 4: Conclusion
The greatest common divisor (GCD) or highest common factor (HCF) of 403, 434, and 465 is **31**. Therefore, the biggest measure that can be used to measure all three quantities exactly is **31 L**.