Okay, let's see. The question is asking for the fifth number in the list of positive integers that have exactly three positive divisors, when they're listed in ascending order. Hmm, I need to figure out what numbers have exactly three divisors and then list them in order until I reach the fifth one.

First, I remember that the number of divisors a number has is related to its prime factorization. For example, if a number is prime, it has exactly two divisors: 1 and itself. But here, we need numbers with exactly three divisors. So, how do numbers get three divisors?

Wait, let me think. If a number has exactly three divisors, then one of them has to be 1, the number itself, and one more. So, the structure of such a number must be such that it's not prime. Because primes have two divisors. So, composite numbers. But which composite numbers have exactly three divisors?

Oh, right! If a number is the square of a prime number, then it has exactly three divisors. Let me check that. Suppose p is a prime, then p² would have divisors 1, p, and p². Yes, that's three divisors. So, the numbers with exactly three divisors are the squares of primes. Therefore, the list should be the squares of primes in ascending order.

So, the first few primes are 2, 3, 5, 7, 11, 13, etc. Squaring them gives 4, 9, 25, 49, 121, 169, and so on. Let me list them in order: 4 (2²), 9 (3²), 25 (5²), 49 (7²), 121 (11²), etc. So, the fifth number in this list would be 121. Wait, but let me confirm if there are any other numbers with exactly three divisors that are not squares of primes. For example, maybe some other composite numbers?

Let me think. Suppose a number n has exactly three divisors. Then the number of divisors formula tells us that if n = p₁^a * p₂^b * ... * p_k^c, then the number of divisors is (a+1)(b+1)...(c+1). So, for the number of divisors to be 3, we need (a+1)(b+1)...(c+1) = 3. Since 3 is prime, the only way this product can be 3 is if there's a single term (a+1) = 3, and all others (if any) are 1. So, n must be p², where p is a prime. Therefore, the only numbers with exactly three divisors are squares of primes. So, there are no other composite numbers with exactly three divisors. Therefore, the list is indeed the squares of primes in ascending order.

Therefore, the first five such numbers would be the squares of the first five primes: 2, 3, 5, 7, 11. Their squares are 4, 9, 25, 49, 121. So, the fifth number is 121. Therefore, the answer should be 121.

But wait, let me double-check. Let's list all numbers starting from 1 and count those with exactly three divisors:

1: divisors 1 (only one divisor)
2: divisors 1,2 (two divisors)
3: divisors 1,3 (two divisors)
4: divisors 1,2,4 (three divisors) → first number
5: divisors 1,5 (two)
6: divisors 1,2,3,6 (four)
7: two divisors
8: divisors 1,2,4,8 (four)
9: divisors 1,3,9 (three divisors) → second number
10: four divisors
11: two
12: six divisors
13: two
14: four
15: four
16: five divisors (1,2,4,8,16)
17: two
18: six
19: two
20: six
21: four
22: four
23: two
24: eight
25: divisors 1,5,25 (three divisors) → third number
26: four
27: four divisors? Wait, 27 is 3³, so divisors are 1,3,9,27 → four divisors. So, not three.
28: six
29: two
30: eight
31: two
32: six
33: four
34: four
35: four
36: nine divisors (1,2,3,4,6,9,12,18,36)
37: two
38: four
39: four
40: eight
41: two
42: eight
43: two
44: six
45: six
46: four
47: two
48: ten divisors
49: divisors 1,7,49 (three divisors) → fourth number
50: six divisors
51: four
52: six
53: two
54: eight
55: four
56: eight
57: four
58: four
59: two
60: twelve divisors
61: two
62: four
63: six
64: seven divisors (1,2,4,8,16,32,64)
65: four
66: eight
67: two
68: six
69: four
70: eight
71: two
72: twelve
73: two
74: four
75: six
76: six
77: four
78: eight
79: two
80: ten divisors
81: divisors 1,3,9,27,81 → five divisors
82: four
83: two
84: twelve divisors
85: four
86: four
87: four
88: eight
89: two
90: twelve divisors
91: four
92: six divisors
93: four
94: four
95: four
96: twelve divisors
97: two
98: six divisors (1,2,7,14,49,98)
99: six divisors
100: nine divisors

Wait, so after 49 (which is 7²), the next number should be 11²=121, right? Let me check numbers between 49 and 121. Let's check 100: divisors as above, which is nine. 101: two divisors. 102: four. 103: two. 104: six. 105: eight. 106: four. 107: two. 108: twelve. 109: two. 110: eight. 111: four. 112: ten. 113: two. 114: eight. 115: four. 116: six. 117: six. 118: four. 119: four. 120: twelve. 121: three divisors (1,11,121). So, after 49 (7²), the next is 121 (11²). Therefore, the list of numbers with exactly three divisors in order is 4,9,25,49,121,... So, fifth number is 121.

Therefore, the answer should be 121. But let me check once again to make sure I haven't missed any numbers between 49 and 121 that might have exactly three divisors. For example, 100 is 10², which is 2²*5², so number of divisors is (2+1)(2+1)=9. 121 is 11², which is prime squared. 125 is 5³, which has divisors 1,5,25,125 → four divisors. 123 is 3*41, which has four divisors. 117 is 3*3*13, which has six divisors. 105 is 3*5*7, which has eight divisors. 115 is 5*23, which has four. 111 is 3*37, four. 105 is already checked. 100 is nine. 99 is 9*11, which is three divisors? Wait, 99 is 9*11, which is 3²*11. So, number of divisors is (2+1)(1+1)=6. So, six divisors. So, 99 has six divisors. 98 is 2*49=2*7², so divisors (1+1)(2+1)=6. 97 is prime. 96 has twelve divisors. 95 is 5*19, four. 94 is 2*47, four. 93 is 3*31, four. 92 is 4*23, six. 91 is 7*13, four. 90 is 2*3²*5, twelve divisors. 89 is prime. 88 is 8*11, six divisors. 87 is 3*29, four. 86 is 2*43, four. 85 is 5*17, four. 84 is 12 divisors. 83 is prime. 82 is 2*41, four. 81 is 3⁴, which has five divisors. 80 is 16*5, ten divisors. 79 is prime. 78 is 6*13, eight. 77 is 7*11, four. 76 is 4*19, six. 75 is 3*5², six. 74 is 2*37, four. 73 is prime. 72 is 12 divisors. 71 is prime. 70 is 2*5*7, eight. 69 is 3*23, four. 68 is 4*17, six. 67 is prime. 66 is 2*3*11, eight. 65 is 5*13, four. 64 is 2⁶, seven divisors. 63 is 7*9=7*3², six. 62 is 2*31, four. 61 is prime. 60 is twelve divisors. 59 is prime. 58 is 2*29, four. 57 is 3*19, four. 56 is 8*7, eight. 55 is 5*11, four. 54 is 6*9=6*3², eight. 53 is prime. 52 is 4*13, six. 51 is 3*17, four. 50 is 2*25=2*5², six. 49 is 7², three divisors. So, after 49, the next number with exactly three divisors is 121. So, the fifth number is 121. Therefore, the answer is 121.
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