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**Answer the questions based on the information given below:**

Mathematicians are assigned a number called Erdos number, (named after the famous mathematician, Paul Erdos). Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of $1.$ For other mathematicians, the calculation of his/her Erdos number is illustrated below:

Suppose that a mathematician X has co-authored papers with several other mathematicians. From among them, mathematician Y has the Smallest Erdos number. Let the Erdos number of Y be $y.$ Then X has the Erdos number of $y+1.$ Hence any mathematician with no co-authorship chain connected to Erdos has an Erdos number of infinity.

- In a seven day long mini-conference organized in memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems. At eh beginning of the conference, A was the only participant who had an infinite Erdos number. Nobody had an Erdos number less than that of F.
- On the third day of the conference F co-authored a paper jointly with A and C. This reduced the average Erdos number of the group of eight mathematicians to $3.$ The Erdos number of B, D, E, G and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdos number of the group of eight to as low as $3.$
- At the end of the third day, five members of this group had identical Erdos numbers while the other three had Erdos numbers distinct from each other.
- On the fifth day, E co-authored a paper with F which reduced the group’s average Erdos number by $0.5.$ The Erdos numbers of the remaining six were unchanged with the writing of this paper.
- No other paper was written during the conference.

The only person having the largest Erdos number at the end of the conference must have had Erdos number (at that time)

- $5$
- $7$
- $9$
- $14$
- $15$