In a group there are `21` childrens, `35` ladies and `49` gents. They all want to stay in same hotel. Find the minimum number of rooms required for their stay so that in each room equal number of members stay with the condition that children, female and male stay in separate rooms.

Answer

To solve the problem of finding the minimum number of rooms required for 21 children, 35 ladies, and 49 gents, where each group must stay in separate rooms and each room must have an equal number of members, we can follow these steps:

### Step 1: Find the factors of each group
We need to find the largest number of members that can stay in each room for each group. This means we need to find the greatest common divisor (GCD) of the number of children, ladies, and gents.

- **Children**: 21
- **Ladies**: 35
- **Gents**: 49

### Step 2: Calculate the GCD
To find the GCD, we can factor each number:

- Factors of 21: \( 1, 3, 7, 21 \)
- Factors of 35: \( 1, 5, 7, 35 \)
- Factors of 49: \( 1, 7, 49 \)

The common factor among these numbers is \( 7 \).

### Step 3: Determine the number of rooms needed for each group
Now that we have the GCD, we can determine how many rooms are needed for each group if 7 members stay in each room.

- For **Children**:
\[ \text{Number of rooms} = \frac{21}{7} = 3 \]

- For **Ladies**:
\[ \text{Number of rooms} = \frac{35}{7} = 5 \]

- For **Gents**:
\[ \text{Number of rooms} = \frac{49}{7} = 7 \]

### Step 4: Calculate the total number of rooms
Now, we can add the number of rooms required for each group to find the total number of rooms needed.

\[ \text{Total rooms} = 3 + 5 + 7 = 15 \]

### Conclusion
The minimum number of rooms required for all members to stay in the hotel, with children, ladies, and gents in separate rooms, is **15**.

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