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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.

How many numbers n < 100 are not divisible by a square of any integer greater than 1?

Find out how many numbers less than 100 are not divisible by a square of any integer greater than 1. This problem uses concepts from number theory, including prime numbers, square-free numbers, and the principle of inclusion-exclusion. Perfect for Grades 9-12.

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