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Consider any number $n$, for which we want to find if it has odd number of factors or not.

Now consider any factor $k$ of $n$, then corresponding to $k$, there will be another factor of $n$ i.e. $n/k$.

So factors exist in pairs. For example, for $n = 18$, the factors pairs are : $(1,18), (2,9), (3,6)$ i.e. 6 factors. We will get odd number of factors only if there is a factor $k$, whose pair is that $k$ only i.e. there exists a $k$ s.t. $n = k*k$ (all other factors have a different corresponding factor). So $n$ should be a perfect square.

So answer would be number of perfect squares less than 100 i.e. 9

Now consider any factor $k$ of $n$, then corresponding to $k$, there will be another factor of $n$ i.e. $n/k$.

So factors exist in pairs. For example, for $n = 18$, the factors pairs are : $(1,18), (2,9), (3,6)$ i.e. 6 factors. We will get odd number of factors only if there is a factor $k$, whose pair is that $k$ only i.e. there exists a $k$ s.t. $n = k*k$ (all other factors have a different corresponding factor). So $n$ should be a perfect square.

So answer would be number of perfect squares less than 100 i.e. 9