Find the greatest possible quantity which can be used to measure exactly the quantities 3L 250mL, 3L 500 mL and 4L

Options

A. 25mL
B. 125mL
C. 250mL
D. 500mL

Answer

To find the greatest possible quantity that can measure exactly the quantities 3L 250mL, 3L 500mL, and 4L, we will follow these steps:

### Step 1: Convert all quantities to milliliters
1. **Convert 3L 250mL to milliliters:**
- 1 liter = 1000 milliliters
- 3 liters = 3 × 1000 = 3000 milliliters
- Therefore, 3L 250mL = 3000 + 250 = **3250 mL**

2. **Convert 3L 500mL to milliliters:**
- 1 liter = 1000 milliliters
- 3 liters = 3 × 1000 = 3000 milliliters
- Therefore, 3L 500mL = 3000 + 500 = **3500 mL**

3. **Convert 4L to milliliters:**
- 1 liter = 1000 milliliters
- 4 liters = 4 × 1000 = **4000 mL**

### Step 2: Find the prime factorization of each quantity
1. **Prime factorization of 3250 mL:**
- 3250 ÷ 2 = 1625
- 1625 ÷ 5 = 325
- 325 ÷ 5 = 65
- 65 ÷ 5 = 13
- 13 ÷ 13 = 1
- Therefore, the prime factorization of 3250 is: **2 × 5^3 × 13**

2. **Prime factorization of 3500 mL:**
- 3500 ÷ 5 = 700
- 700 ÷ 5 = 140
- 140 ÷ 5 = 28
- 28 ÷ 7 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
- Therefore, the prime factorization of 3500 is: **5^3 × 7 × 2^2**

3. **Prime factorization of 4000 mL:**
- 4000 ÷ 5 = 800
- 800 ÷ 5 = 160
- 160 ÷ 5 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
- Therefore, the prime factorization of 4000 is: **5^3 × 2^5**

### Step 3: Identify the common prime factors
- The common prime factors from the factorizations:
- For 3250: **2 × 5^3 × 13**
- For 3500: **5^3 × 7 × 2^2**
- For 4000: **5^3 × 2^5**

### Step 4: Find the highest common factor (HCF)
- The common prime factor is **5^3** (which is 125).
- The common factor for 2 is the lowest power, which is **2^1** (which is 2).
- Therefore, the HCF = 5^3 × 2^1 = 125 × 2 = **250 mL**.

### Final Answer:
The greatest possible quantity that can measure exactly the quantities 3L 250mL, 3L 500mL, and 4L is **250 mL**.

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