300 views
How many number less than 100 has an odd number of factors ??

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
Actually this is just a pattern . You need to know this that numbers with odd number of factors are perfect squares .

1,4,9,16,25,36,49,64,81 these are the number having odd number of factors .  Ans is : 9

To find number of factors :

49 = 7^2

No of factors = (2+1) = 3 which are 1,7,49 .

I think 9 should be the ans..Bcz All the perfect square have odd number of factors and as they told <100 so we can't consider 100 So ans is 9.

No.   Odd no. Of Factors

1      1

4.     1,2,4

9.     1,3,9

16.   1,2,4,8,16 .

likewise .

81   1,3,9,27,81

Correct me if I'm wrong :)