<Question>
Each week, between 30 and 50 students show up for an archery class run by Betty and Wilma.  Usually the students break up into groups of equal size for target practice.  However, this week, Betty noticed that she could not break the students up into multiple groups of equal size.  Wilma noticed that if she and Betty both joined the students in practicing, they still could not break the archers up into groups of equal size.  How many students showed up to the archery class this week?
</Question>

<think>
<Plan_1>
Okay, let's try to figure out this archery class problem. So, the question is about finding the number of students that showed up this week, given certain conditions about grouping. Let me break it down step by step.

<prompt_1.1> Summarize the problem and identify the mathematical concepts involved. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, Betty and Wilma are running an archery class, and usually, the students break into groups of equal size for target practice. But this week, Betty noticed that she couldn't split the students into multiple groups of equal size. Then Wilma noticed that even if both of them joined the students, they still couldn't break them into groups of equal size. The number of students this week was between 30 and 50. We need to find that number.

Hmm. Let's think. So, the key here is about divisibility. If they can't break into multiple groups of equal size, that means the number of students is a prime number, right? Because prime numbers can only be divided into groups of 1 and themselves. But wait, the problem says "multiple groups," so if the number is prime, the only possible groupings are 1 group or the number itself, which isn't multiple groups. So, maybe the number of students is a prime number. But Betty noticed that she (possibly alone) couldn't break them into multiple groups. Then Wilma joining (so total students +2) also can't be broken into multiple groups. So, both N and N+2 are primes? Wait, but primes differ by 2 are called twin primes. But the number of students is between 30 and 50. Let's check if there are any twin primes in that range.

Wait, but let me confirm. If Betty is the one who noticed she can't break the students into multiple groups, that would mean that the number of students is not divisible by any number other than 1 and itself. So, if Betty is a single person, maybe she tried to split the students into groups where each group has more than one person, but they can't. So, the number of students must be a prime number. Then, when Wilma joins, making it N+2 students (since Wilma and Betty are two more people), they still can't be split into multiple groups, so N+2 is also prime. Therefore, we need two primes that are two apart, i.e.

 twin primes. So, let's list primes between 30 and 50. Let me recall the primes in that range: 31, 37, 41, 43, 47. Now, check for twin primes. 31 and 37 are 6 apart. 37 and 41 are 4 apart. 41 and 43 are 2 apart. So, 41 and 43 are twin primes. So, if N=41, then N+2=43. Both are primes.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Therefore, the number of students this week is 41. Because 41 is a prime number, so Betty couldn't split 41 students into multiple groups (since 41 is prime, only 1 and 41). Then, when Wilma joins, making it 43 students, which is also prime, so they still can't split into multiple groups.

But wait, let me check again. The problem states "between 30 and 50 students". So, 30 < N < 50. The number N is between 30 and 50. So, if N=41, that's between 30 and 50.

<prompt_2.1> Verify the solution by checking if 41 and 43 are primes and fit the problem's conditions. </prompt_2.1>

But let me verify again. Let's check if there are any other possibilities.

<prompt_2.2> Check for other possibilities of twin primes in the range and confirm the solution. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Then N+2=43, which is also less than 50. Wait, 43 is less than 50? Wait, 43 is less than 50, yes. Wait, 43 is 43, which is less than 50. So, that works. So, 41 and 43 are both primes. So, N=41. But let me check if there are other possibilities. Let's see, the primes between 30 and 50 are 31, 37, 41, 43, 47. Checking the differences: 31 and 37 differ by 6, so not twin primes. 37 and 41 differ by 4, 41 and 43 differ by 2, 43 and 47 differ by 4. Then 47 and next prime is 53, which is over 50. So, the only twin primes in that range are 41 and 43. Therefore, N=41.
</execution_2.1>

<execution_2.2>
For example, 31 and 33. But 33 is not prime. 37 and 39. 39 is not prime. 43 and 45. 45 is not prime. So, the only pair in the range is 41 and 43. So, N=41.

But wait, another thought. The problem says "multiple groups of equal size". So, if the number is a prime, then the only possible group sizes are 1 and the prime itself. So, if you can't form multiple groups, that means you can't form groups of size greater than 1 and less than the total. So, yes, primes fit. But is there another possibility? For example, maybe the number is a prime power? Wait, no. If it's a prime power, say p^k, then it can be divided into p groups of size p^(k-1), etc. So, unless the number is a prime, it might have factors. Wait, but the problem states that Betty couldn't break them into multiple groups. So, if the number is composite, she should be able to, right? So, if she can't, then the number must be prime. Therefore, N is prime, and N+2 is also prime. Therefore, twin primes.

So, in the range 30-50, the only such number is 41. Because 41 is prime, 43 is prime. Let's check if there are others. Next twin primes after 41 are 59 and 61, which are outside the range. Before 41, 31 and 37. 31 is prime, 37 is prime, but 31 +2 is 33, which is not prime. Wait, no. Wait, if N=31, then N+2=33. But 33 is not prime. So, that doesn't work. Similarly, 43 +2=45, which is not prime. So, only 41 and 43 work.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
Therefore, the number of students is 41. Let me check once again.

<prompt_3.1> Confirm the solution by checking the conditions with N=41 and verifying prime numbers. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
If there are 41 students, Betty tries to split them into groups. Since 41 is prime, she can't split them into groups of equal size unless she has 1 group or 41 groups of 1. But the problem says "multiple groups," so more than one group. So, 41 can't be split into multiple groups. Then, when Wilma joins, making it 43 students. 43 is also prime, so same reasoning. Therefore, 41 is the answer.
</execution_3.1>
</Execution_3>

<Final_answer>
Thus, the number of students who showed up to the archery class this week is $\boxed{41}$.
</Final_answer>
</think>