Okay, let's try to figure out how many students showed up to the archery class this week. The problem says that usually, the students break up into groups of equal size, but this week Betty noticed they couldn't. Then Wilma noticed that even if both of them joined, they still couldn't break them into equal groups. The number of students is between 30 and 50. Hmm, let's break this down step by step.

First, let's understand what it means for students to be able to break into groups of equal size. That means the number of students must be divisible by some number greater than 1. So, if they can't break into equal groups, the number of students must be a prime number or have very limited divisors. Wait, but the problem says they usually can break into groups, so the usual number is composite, but this week it's prime or something. Wait, no, the usual number is between 30 and 50, but this week they couldn't. So this week's number is a number between 30 and 50 that isn't divisible by any number other than 1 and itself? Or maybe it's a number that only has a few divisors. Wait, but even prime numbers have two divisors, so if they couldn't break into multiple groups, maybe the number of students this week is a prime number? But then, if they couldn't break into groups of equal size, that would mean the number is prime, since primes can't be divided into equal groups except for groups of 1 or themselves. But then, if that's the case, then this week's number is a prime number between 30 and 50. Let's list those primes: 31, 37, 41, 43, 47. But then Wilma noticed that even if both Betty and Wilma joined, which would add 2 more students, making it n + 2. So n + 2 also can't be broken into equal groups. But if n is prime, n + 2 could be composite. For example, 31 + 2 = 33, which is divisible by 3 and 11. So 33 can be divided into groups of 3, 11, etc. But the problem says even with both joining, they still couldn't break into equal groups. So n + 2 must also be a number that can't be divided into equal groups, meaning it's also prime? But 31 + 2 is 33, which is not prime. So that doesn't work. Therefore, maybe n isn't prime. Hmm, so perhaps n is a composite number that is not divisible by any number other than 1 and itself, but that's a contradiction. Wait, maybe the number is such that it's not divisible by any number except 1 and itself, but that's a prime. So maybe the number is prime, but then n + 2 is also prime? Let's check the primes between 30 and 50. Let's list them again: 31, 37, 41, 43, 47. Now, adding 2: 33, 39, 43, 45, 49. Of these, 43 and 49 are primes. Wait, 43 is prime, 49 is 7 squared. So if n is 43, then n + 2 is 45, which is 9*5, so divisible by 3, 5, etc. So 45 can be divided into groups of 3, 5, etc. But the problem states that even with both joining, they couldn't break into equal groups. So 45 can be divided into equal groups, so that's not acceptable. Similarly, 49 is 7*7, so 49 can be divided into 7 groups of 7. So that's not acceptable. So 43 + 2 = 45, which can be divided, so that's not good. Similarly, 37 + 2 = 39, which is 3*13, so divisible. 31 + 2 = 33, which is 3*11. 41 + 2 = 43, which is prime. So if n is 41, then n + 2 is 43, which is prime. So in that case, n is 41, which is prime, n + 2 is 43, which is also prime. So then, both n and n + 2 are primes. So n is a prime, and n + 2 is also a prime. These are called twin primes. So 41 and 43 are twin primes. So maybe this is the case. Let me check if 41 is between 30 and 50. Yes, 41 is. Then 43 is also between 30 and 50. So if n is 41, then this week there are 41 students. They can't be divided into multiple groups of equal size except groups of 1 and 41. Then, if both Betty and Wilma join, the total is 43, which is also prime, so they can't be divided into equal groups except 1 and 43. So that fits the problem's description. So the answer is 41. But let me check other possibilities.

Wait, but the problem says "groups of equal size" for target practice. So if the number is 41, then unless there's a group size of 41, they can't form multiple groups. So that's correct. Then adding 2 makes 43, which is also a prime. So that works. But let's check other primes. For example, 37: 37 + 2 = 39, which is 3*13. So 39 can be divided into groups of 3 or 13. So that's not acceptable. Similarly, 31 + 2 = 33, which can be divided. 43 + 2 = 45, which can be divided. So only 41 and 43 are twin primes in that range. So n is 41. So the answer is 41. But let me think again. Another way to approach this is to find numbers between 30 and 50 that are primes, and when you add 2, the result is also a prime. So twin primes in that range. Twin primes are pairs of primes that are two apart. So 31 and 33 (33 not prime), 37 and 39 (39 not prime), 41 and 43 (both primes), 43 and 45 (45 not prime), 47 and 49 (49 not prime). So only 41 and 43 are twin primes. Therefore, n must be 41. So the answer is 41. Let me verify once more. If there are 41 students, they can't form multiple groups (since 41 is prime), so they can only form 41 groups of 1 or 1 group of 41. Then, if Betty and Wilma join, total is 43, which is also prime. So they can't form multiple groups. So that fits the problem. Therefore, the answer is 41. But wait, the problem says "between 30 and 50", so 30 to 50 inclusive. 41 is within that range. So that's correct. Therefore, the answer is 41. So I think that's the solution.
Therefore, the number of students that showed up this week is $\boxed{41}$.
</think>