<s>[INST] Number of squarefree positive integers less than $100$ 
An integer is called squarefree if it is not divisible by the square
  of a positive integer greater than $1$. Find the number of squarefree
  positive integers less than $100$.

My attempt:
I apply the inclusion-exclusion principle directly.


*

*Total number of integers = 99

*Number of integers divisible by $2^2$ = $24$

*Number of integers divisible by $3^2$ = $11$

*Number of integers divisible by $4^2$ = $6$

*Number of integers divisible by $5^2$ = $3$

*Number of integers divisible by $6^2$ = $2$

*Number of integers divisible by $7^2$ = $2$

*Number of integers divisible by $8^2$ = $1$

*Number of integers divisible by $9^2$ = $1$

*Number of integers divisible by $2^2$ and $3^2$ = $2$

*Number of integers divisible by $2^2$ and $4^2$ = $1$


Then the required solution would be $99-(24+11+6+3+2+2+1+1)+2+1=52$. But the solution is $61$. Where is my mistake?
 [/INST] Something that may come in handy if you don't need the exact number of squarefree positive integers equal to or below $n$ and an approximation is sufficient: if we denote by $S(n)$ this quantity, we have:
$$
S(n) \approx \frac{n}{\zeta(2)},
$$
where $\zeta$ denotes the Riemann zeta function and 
$$
\zeta(n) = \frac{\pi^2}{6}  \approx 1.64493406684822\ldots
$$
So in your case the number of positive integers that are squarefree below $100$ is 
$$
\frac{100}{\zeta(2)} \approx 60.7927101\ldots,
$$
which is not far from the actual value of $61$.
</s>