# The Brilliant Followers

Let there be $$100$$ brilliant members labelled $$a_1, a_2, a_3, a_4, a_5,\ldots, a_{99}, a_{100}$$ In this group of brilliant members, $$a_2$$ follows $$a_1$$, $$a_3$$ follows $$a_2$$, $$\ldots$$ $$a_n$$ follows $$a_{n-1}$$, till $$a_{100}$$ follows $$a_{99}$$. Other than this group of members, these members have other followers, denoted by $$F$$ who themselves have no followers. Now, $$a_n$$ has $$n$$ followers. Thus, for $$a_1$$, the only follower is $$a_2$$, while for $$a_2$$, he has two followers: $$a_3$$ and $$1 F$$. This pattern continues till $$a_{100}$$.

Now, say $$a_1$$ makes a post, the total number of possible reshares which can be done by $$a_2, a_3, a_4,\ldots a_{99}, a_{100}$$ and the $$F$$s is given by $$x$$. All of the $$F$$s will definitely reshare a post made or reshared by the person(s) they follow, and all $$a_{2n+1}$$, with $$n\in\mathbb{N}$$, will do the same thing. However, for all $$a_{2n}$$, with $$n\in\mathbb{N}$$, the probability that they will reshare something posted by someone they follow is $$\frac12$$. The probability that the number of reshares is $$\geq\frac{x}{2}$$ is given by $$\frac{1}{a}$$. Find the last digit of $$a$$

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