The question asks how many numbers $n < 100$ are not divisible by a square of any integer greater than 1.

This means we are looking for numbers that are square-free, i.e., numbers that are not divisible by the square of any prime number (such as $4 = 2^2$, $9 = 3^2$, $25 = 5^2$, etc.).

Steps to solve:

1. Prime squares up to 100: The squares of primes that are less than or equal to $100$ are:

   • $2^2 = 4$
   • $3^2 = 9$
   • $5^2 = 25$
   • $7^2 = 49$

2. Numbers divisible by squares:

   • Numbers divisible by $4$ (i.e., $2^2$) are $4, 8, 12, \dots, 96$. There are $\left\lfloor \frac{99}{4} \right\rfloor = 24$ such numbers.
   • Numbers divisible by $9$ (i.e., $3^2$) are $9, 18, 27, \dots, 99$. There are $\left\lfloor \frac{99}{9} \right\rfloor = 11$ such numbers.
   • Numbers divisible by $25$ (i.e., $5^2$) are $25, 50, 75$. There are 3 such numbers.
   • Numbers divisible by $49$ (i.e., $7^2$) are $49, 98$. There are 2 such numbers.

3. Using the principle of inclusion-exclusion, we calculate how many numbers are divisible by these squares.

   • First, sum the counts of numbers divisible by $4, 9, 25,$ and $49$: $24 + 11 + 3 + 2 = 40$.
   • Next, subtract the overlap (numbers divisible by more than one square):
     • Numbers divisible by $4 \times 9 = 36$ are $36, 72$. There are 2 such numbers.
     • Numbers divisible by $4 \times 25 = 100$ (but $100$ is outside our range).
     • Numbers divisible by $4 \times 49 = 196$ (also outside our range).
     • Numbers divisible by $9 \times 25 = 225$ (outside our range).
     • No overlap among other pairs.

   So, the total number of numbers divisible by the square of any prime is: $40 - 2 = 38$

4. Square-free numbers: There are $99$ numbers below $100$ (from $1$ to $99$). The number of square-free numbers is the complement: $99 - 38 = 61$

Thus, 61 numbers less than 100 are not divisible by a square of any integer greater than 1.

Would you like more details on any step? Here are 5 related questions to further explore this concept:

1. How do you compute square-free numbers in a larger range?
2. Can you explain the principle of inclusion-exclusion in more detail?
3. What are square-free numbers used for in number theory?
4. How would the result change if we included higher prime squares like $11^2$?
5. Can we generalize this method for any upper limit, not just 100?

Tip: To find square-free numbers, it's helpful to focus on prime numbers and their powers because composite numbers derive from these primes.