How many odd numbers between 10 and 1,000 are the squares of integers?

1. Translate the problem requirements: Find perfect squares (numbers that equal some integer squared) that are odd numbers and fall between 10 and 1,000
2. Establish the range of base integers: Determine which integers, when squared, produce results between 10 and 1,000
3. Apply odd/even properties: Use the principle that only odd integers produce odd squares to filter our candidates
4. Count the qualifying integers: Count how many odd integers fall within our established range

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the question is asking in everyday terms. We need to find numbers that satisfy three conditions at the same time:

First, the number must be a "square of an integer" - this means it's a perfect square. For example, 16 is a perfect square because \(4 \times 4 = 16\). We say "4 squared equals 16" or write it as \(4^2 = 16\).

Second, the number must be odd. Remember that odd numbers end in 1, 3, 5, 7, or 9.

Third, the number must fall between 10 and 1,000. This means we want numbers greater than 10 but less than 1,000.

So we're looking for perfect squares that are odd and fall in the range from 11 to 999.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical requirements

2. Establish the range of base integers

Now we need to figure out which integers, when squared, give us results between 10 and 1,000.

Let's think about this step by step. If we square an integer and want the result to be greater than 10, what's the smallest integer we need to consider?

Let's check: \(3^2 = 9\) (too small, since 9 < 10)
\(4^2 = 16\) (this works, since 16 > 10)

So we need integers of at least 4.

Now for the upper limit. If we square an integer and want the result to be less than 1,000, what's the largest integer we can use?

Let's check: \(31^2 = 961\) (this works, since 961 < 1,000)
\(32^2 = 1,024\) (too big, since 1,024 > 1,000)

So we need integers of at most 31.

This means we're looking at integers from 4 to 31 when we square them.

3. Apply odd/even properties

Here's a key insight that makes this problem much simpler: only odd integers produce odd squares.

Let me explain why this is true. When you square an even number, you get an even result. When you square an odd number, you get an odd result.

For example:
• Even: \(4^2 = 16\) (even result)
• Odd: \(5^2 = 25\) (odd result)
• Even: \(6^2 = 36\) (even result)
• Odd: \(7^2 = 49\) (odd result)

Since we only want odd perfect squares, we only need to consider odd integers in our range from 4 to 31.

The odd integers from 4 to 31 are: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31

4. Count the qualifying integers

Let's count the odd integers from our list and verify a few of their squares fall in our desired range:

The odd integers are: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31

Let's verify the boundary cases:
• \(5^2 = 25\) (✓ between 10 and 1,000, and odd)
• \(31^2 = 961\) (✓ between 10 and 1,000, and odd)

Counting our list: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31

That's 14 odd integers.

Process Skill: APPLY CONSTRAINTS - Using the odd/even property to filter our candidates efficiently

Final Answer

There are 14 odd numbers between 10 and 1,000 that are squares of integers.

The answer is C. 14.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the range constraints
Students often include the boundary values 10 and 1,000 in their search, when the problem asks for numbers "between" these values. This means they should be looking for numbers from 11 to 999, not 10 to 1,000. Including 10 would mean considering whether 10 is an odd perfect square (which it isn't), but the conceptual error of boundary inclusion can lead to other mistakes.

Faltering Point 2: Not recognizing the odd integer connection
Many students start by listing all perfect squares in the range and then checking which ones are odd, rather than realizing that only odd integers produce odd squares. This leads to a much longer, more error-prone approach where they might miss some squares or miscount.

Faltering Point 3: Incorrect range determination for base integers
Students may incorrectly determine which integers to square. They might think that since we want results greater than 10, we start with integers greater than 10, not realizing we need to find which integer, when squared, first exceeds 10. Similarly for the upper bound with 1,000.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in boundary calculations
When determining that \(31^2 = 961\) and \(32^2 = 1,024\), students often make calculation mistakes, especially with larger squares. These errors can shift their entire range of qualifying integers, leading to incorrect counts.

Faltering Point 2: Missing integers when listing odd numbers
When listing odd integers from 5 to 31, students frequently skip numbers (like forgetting 21 or 27) or accidentally include even numbers. This systematic listing requires careful attention that students often rush through.

Faltering Point 3: Incorrectly determining the starting point
Students might start their count from 3 instead of 5, not properly verifying that \(3^2 = 9\) is less than 10. This adds an extra integer to their count, leading them to answer 15 instead of 14.

Errors while selecting the answer

No likely faltering points