<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] There are 6 integers divisible by 2^2 and 4^2, not 1. You are also not counting when integers are divisible by both 3^2 and 9^2, or 2^2, 4^2, and 8^2 for example. The other methods provided are easier to calculate, but your approach is correct except for these errors.
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