### Component 1
<description>
Initial understanding and approach to solve the problem of finding numbers with an even number of divisors.
</description>

<plan>
Okay, let me try to figure out how many positive integers less than 60 have an even number of positive divisors. Hmm, so first, I remember that most numbers have an even number of divisors because divisors come in pairs. For example, if a number has a divisor d, it also has the divisor n/d. So that would mean divisors come in pairs, right? But wait, there's an exception when a number is a perfect square. Because if a number is a square, like 4, one of its divisors is the square root, which is repeated. So for 4, the divisors are 1, 2, 4. Here, 2 is the square root and only counts once. So the number of divisors is odd for perfect squares. Therefore, non-square numbers should have an even number of divisors. So, if I want to find numbers less than 60 with an even number of divisors, I need to exclude the perfect squares.
</plan>

<prompt>
Based on the above plan, calculate the total number of integers less than 60 and determine how to exclude perfect squares to find numbers with an even number of divisors.
</prompt>

<execution>
So, the total numbers from 1 to 59 are 59 numbers. Then, subtract the number of perfect squares in that range. That should give the count of numbers with even number of divisors.
</execution>


### Component 2
<description>
Verification of the understanding that perfect squares have an odd number of divisors.
</description>

<plan>
Wait, but let me verify this. Let's take an example.
</plan>

<prompt>
Verify the understanding that perfect squares have an odd number of divisors by checking examples.
</prompt>

<execution>
Take number 6. Its divisors are 1, 2, 3, 6. That's 4 divisors, which is even. Number 4: 1, 2, 4. Three divisors, odd.
</execution>


### Component 3
<description>
Calculation of non-perfect square numbers less than 60.
</description>

<plan>
So the problem reduces to finding how many numbers less than 60 are not perfect squares. So total numbers from 1 to 59: 59 numbers. Number of perfect squares between 1 and 59: let's see.
</plan>

<prompt>
Calculate the number of perfect squares less than 60 and subtract from the total numbers to find non-perfect squares.
</prompt>

<execution>
The squares are 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49, 8^2=64. But 64 is over 60, so up to 7^2=49. So that's 7 perfect squares. Therefore, the number of non-perfect squares is 59 - 7 = 52.
</execution>


### Component 4
<description>
Re-evaluation of the inclusion of 1 as a perfect square and recalculation.
</description>

<plan>
Wait, but hold on. Wait, the problem says "positive integers less than 60". Does that include 1? Because 1 is a positive integer less than 60. But 1 is a perfect square (1^2=1), so it's included in the count of perfect squares.
</plan>

<prompt>
Re-evaluate the inclusion of 1 as a perfect square and recalculate the number of non-perfect squares.
</prompt>

<execution>
So total perfect squares from 1 to 59 are 1,4,9,16,25,36,49. That's 7 numbers. So total numbers with even number of divisors would be 59 - 7 = 52.
</execution>


### Component 5
<description>
Verification of the formula with a smaller range of numbers.
</description>

<plan>
But let me check another example. Let's take numbers from 1 to 10. There are 10 numbers.
</plan>

<prompt>
Verify the formula by applying it to numbers from 1 to 10 and list numbers with even divisors.
</prompt>

<execution>
Perfect squares are 1,4,9. So 3 numbers. Therefore, numbers with even divisors: 10 - 3 = 7. Let's list them: 2,3,5,6,7,8,10. Each of these should have even number of divisors. For 2: divisors 1,2 (even). 3: 1,3 (even). 5: 1,5 (even). 6: 1,2,3,6 (even). 7: 1,7 (even). 8: 1,2,4,8 (even). 10: 1,2,5,10 (even).
</execution>


### Component 6
<description>
Reaffirmation of the solution using the formula for numbers less than 60.
</description>

<plan>
So applying the same logic, 59 total numbers minus 7 perfect squares gives 52. But wait, hold on. Wait, the question is "positive integers less than 60", which would be numbers from 1 to 59 inclusive.
</plan>

<prompt>
Reaffirm the solution using the formula for numbers less than 60 and calculate the result.
</prompt>

<execution>
So that's 59 numbers. Perfect squares in this range: 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49. The next square is 8^2=64, which is more than 59, so we have 7 perfect squares. So 59 - 7 = 52.
</execution>


### Component 7
<description>
Manual recount of perfect squares to ensure accuracy.
</description>

<plan>
But wait, let me check again. Let me count the perfect squares manually: 1,4,9,16,25,36,49.
</plan>

<prompt>
Manually recount the perfect squares less than 60 to ensure accuracy.
</prompt>

<execution>
That's 7 numbers. So 59 total numbers minus 7 gives 52 numbers with even number of divisors.
</execution>


### Component 8
<description>
Final confirmation of the count of perfect squares and the solution.
</description>

<plan>
Hmm, but let me just make sure that I didn't make a mistake in the count of perfect squares. Let's list them: n=1: 1^2=1, yes.
</plan>

<prompt>
Confirm the count of perfect squares and the solution by listing them again.
</prompt>

<execution>
n=2: 4, yes. n=3:9, n=4:16, n=5:25, n=6:36, n=7:49. n=8:64 which is over 60. So that's 7. Correct. So 59 -7=52.
</execution>


### Component 9
<description>
Final answer and additional confirmation with another approach.
</description>

<plan>
Therefore, the number of positive integers less than 60 with an even number of positive divisors is 52. Wait, but let me confirm once again with another approach.
</plan>

<prompt>
Confirm the final answer with another approach by checking a specific number.
</prompt>

<execution>
Let me pick a number, say 16. Divisors of 16 are 1,2,4,8,16. That's 5 divisors, which is odd. So 16 is a perfect square and is excluded. So non-perfect squares should have even divisors. So yes, 59 total, subtract 7 perfect squares gives 52.
</execution>
