<Question>
How many positive integers less than 60 have an even number of positive divisors?
</Question>

<Plan_1>
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.

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.

Wait, but let me verify this. Let's take an example.

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.

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.

<prompt_1.1> Calculate the number of perfect squares less than 60 and subtract from the total numbers to find non-perfect squares. </prompt_1.1>
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But let me check another example. Let's take numbers from 1 to 10. There are 10 numbers.

<prompt_1.2> Verify the formula by applying it to numbers from 1 to 10 and list numbers with even divisors. </prompt_1.2>
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But wait, let me check again. Let me count the perfect squares manually: 1,4,9,16,25,36,49.

<prompt_1.3> Manually recount the perfect squares less than 60 to ensure accuracy. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
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.

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.

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_1.1>

<execution_1.2>
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_1.2>

<execution_1.3>
That's 7 numbers. So 59 total numbers minus 7 gives 52 numbers with even number of divisors.

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.

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_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
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.

<prompt_2.1> Reaffirm the solution using the formula for numbers less than 60 and calculate the result. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
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.

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.

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_2.1>
</Execution_2>

<Final_answer>
Therefore, there are $\boxed{52}$ positive integers less than 60 that have an even number of positive divisors.
</Final_answer>