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At a management school, the oldest $10$ dorms, numbered $1$ to $10$, need to be repaired urgently, The following diagram represents the estimated repair costs (in Rs. Crores) for the $10$ dorms. For any dorm, the estimated repair cost (in Rs. Crores) is an integer. Repairs with estimated cost Rs. $1$ or $2$ Crores are considered light repairs, repairs with estimated cost Rs. $3$ or $4$ are considered moderate repairs and repairs with estimated cost Rs. $5$ or $6$ Crores are considered extensive repairs.

Further, the following are known:

- Odd-numbered dorms do not need light repair, even-numbered dorms do not need moderate repair and dorms, whose numbers are divisible by $3$, do not need extensive repair.
- Dorms $4$ to $9$ all need different repair costs, with Dorm $7$ needing the maximum and Dorm $8$ needing the minimum.

Suppose further that:

- $4$ of the $10$ dorms needing repair are women's dorms and need a total of Rs. $20$.Crores for repair.
- Only one of Dorms $1$ to $5$ is a women's dorm.

Which of the following is a women's dorm?

- Dorm $2$
- Dorm $5$
- Dorm $8$
- Dorm $10$

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