What is the number of non-square natural numbers between 10² and 12² 

Let's find the number of non-square natural numbers between \( 10^2 \) and \( 12^2 \).

1. **Calculate the squares**:
   \[
   10^2 = 100
   \]
   \[
   12^2 = 144
   \]

2. **List the natural numbers between 100 and 144**:
   The natural numbers between 100 and 144 are:
   \[
   101, 102, 103, \ldots, 143
   \]
   The total number of numbers is \( 144 - 100 - 1 = 43 \).

3. **Identify the perfect squares in this range**:
   Let's check for any perfect squares between 100 and 144.

   - \( 11^2 = 121 \)

   So, 121 is the only perfect square between 100 and 144.

4. **Count the non-square numbers**:
   Since there is one perfect square (121) between 100 and 144, the number of non-square natural numbers is:
   \[
   43 - 1 = 42
   \]

Thus, the number of non-square natural numbers between \( 10^2 \) and \( 12^2 \) is \( 42 \).

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How many non-square natural numbers lie between \( 12^2 \) and \( 15^2 \)?
2. What is the number of perfect squares between \( 5^2 \) and \( 9^2 \)?
3. How many non-square natural numbers are there between \( 15^2 \) and \( 18^2 \)?
4. What is the number of non-square natural numbers between \( 8^2 \) and \( 10^2 \)?
5. Can you determine the total number of natural numbers between \( 20^2 \) and \( 25^2 \)?

**Tip**: When counting non-square numbers between \( n^2 \) and \( m^2 \), consider checking for squares of integers within the interval \((n, m)\) to find the non-square count efficiently.

Discover the number of non-square natural numbers between 10² and 12² using mathematical analysis. Learn about perfect squares, identify natural numbers, and count non-square numbers efficiently.

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