A milk seller has three types 403,465 and 496 gallon of milk. Find the minimum number of bottles, which can measure the whole quantity of milk without mixing them ?

Options

A. 34
B. 46
C. 26
D. 44

Answer

To solve the problem of finding the minimum number of bottles that can measure the whole quantity of milk without mixing them, we will follow these steps:

### Step 1: Identify the quantities of milk
The quantities of milk given are:
- 403 gallons
- 465 gallons
- 496 gallons

### Step 2: Find the Highest Common Factor (HCF)
To find the HCF of the three quantities, we will factor each number:

1. **Factor 403**:
- 403 can be factored as \(13 \times 31\).

2. **Factor 465**:
- 465 can be factored as \(15 \times 31\) (or \(3 \times 5 \times 31\)).

3. **Factor 496**:
- 496 can be factored as \(16 \times 31\) (or \(2^4 \times 31\)).

From the factorizations, we can see that the common factor among all three quantities is \(31\).

### Step 3: Calculate the HCF
Thus, the HCF of 403, 465, and 496 is:
\[ \text{HCF} = 31 \]

### Step 4: Determine the number of bottles for each type of milk
Now, we will calculate how many bottles of 31 gallons are needed for each quantity of milk:

1. **For 403 gallons**:
\[ \text{Number of bottles} = \frac{403}{31} = 13 \]

2. **For 465 gallons**:
\[ \text{Number of bottles} = \frac{465}{31} = 15 \]

3. **For 496 gallons**:
\[ \text{Number of bottles} = \frac{496}{31} = 16 \]

### Step 5: Calculate the total number of bottles
Now, we sum the number of bottles required for each type of milk:
\[ \text{Total number of bottles} = 13 + 15 + 16 = 44 \]

### Final Answer
The minimum number of bottles required to measure the whole quantity of milk without mixing them is:
\[ \boxed{44} \]

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