A person has to completely put each of three types of juices, 210 litres of orange juice, 220 litres of guava juice and 260 litres of litchi juice in bottles of equal size without mixing any of the above three types of juices such that each bottle is completely filled. What is the least possible number of bottles required?

Options

A. 45
B. 69
C. 72
D. 55

Answer

To solve the problem of how many bottles are needed to store the juices without mixing them, we will follow these steps:

### Step 1: Identify the quantities of each type of juice
- Orange juice: 210 liters
- Guava juice: 220 liters
- Litchi juice: 260 liters

### Step 2: Find the highest common factor (HCF) of the three quantities
To find the HCF, we need to factor each quantity into its prime factors.

1. **Factor 210**:
- 210 = 2 × 3 × 5 × 7

2. **Factor 220**:
- 220 = 2 × 2 × 5 × 11 = 2² × 5 × 11

3. **Factor 260**:
- 260 = 2 × 2 × 5 × 13 = 2² × 5 × 13

### Step 3: Determine the common factors
The common prime factors from the factorizations are:
- The only common prime factor among all three quantities is 10 (which is 2 × 5).

### Step 4: Calculate the number of bottles needed for each type of juice
Now that we have determined that the bottle size should be 10 liters, we can calculate how many bottles are needed for each type of juice:

1. **For Orange Juice**:
- Number of bottles = Total liters of orange juice / Size of each bottle
- Number of bottles = 210 liters / 10 liters = 21 bottles

2. **For Guava Juice**:
- Number of bottles = Total liters of guava juice / Size of each bottle
- Number of bottles = 220 liters / 10 liters = 22 bottles

3. **For Litchi Juice**:
- Number of bottles = Total liters of litchi juice / Size of each bottle
- Number of bottles = 260 liters / 10 liters = 26 bottles

### Step 5: Calculate the total number of bottles required
Now, we add the number of bottles required for each type of juice:
- Total bottles = Bottles for orange juice + Bottles for guava juice + Bottles for litchi juice
- Total bottles = 21 + 22 + 26 = 69 bottles

### Final Answer:
The least possible number of bottles required is **69 bottles**.

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