# How many natural numbers less than 200 will have 12 factors/divisors?

How many natural numbers less than 200 will have 12 factors (a.k.a. divisors)?

I think the answer is $11$.

Firstly there can be at most $3$ distinct prime factors.

$12=1\cdot12 =2\cdot6 =3\cdot4 =2\cdot2\cdot3$

$N=a^{11} =a\cdot b^5 =a^2\cdot b^3 =a\cdot b\cdot c^2$

Then, $1$ prime factor is not possible because the smallest $2^{11}> 200$. So, the options are:

For $2$ prime factors:
$72, 96, 160$

For $3$ prime factors:
$60,90,150,84,140,126,132,156$.

I think this list is exhaustive. But am I right? Is there any other approach?

• Why did you stop formatting in MathJax in the middle of the question? Also, use \cdot to write the multiplication dot. – 5xum Jan 7 '15 at 9:20
• I don't understand the question, could you please explain better what are you looking for? For me $2$ is the smallest possible factor in $\Bbb{Z}$, but $2^{12}=4096 > 200$... – Bman72 Jan 7 '15 at 9:20
• @Ale Say 72=2^3*3^2. Then number of factors of 72=(3+1)*(2+1)=12 – archangel89 Jan 7 '15 at 9:25
• @Ale The question is clearly posed given the definition of a factor. For example, $72$ has $12$ factors: $1,2,3,4,6,8,9,12,18,24,36,72$ – Zubin Mukerjee Jan 7 '15 at 9:26
• @ZubinMukerjee My bad. I swapped factors with irreducible. Now is clear, thank you. – Bman72 Jan 7 '15 at 9:28

Hint: there are necessarily two factors $a,b$ with $\gcd(a,b) = 1$. Then using the fact that $\tau$ is multiplicative, you must have $\tau(a)\tau(b) = 12$ with $\min(\tau(a), \tau(b)) > 1$. This much simplifies the problem.

• Can you please elaborate a little more? It may sound silly to you but I could not quite understand the hint. – archangel89 Jan 7 '15 at 10:41
• the solutions have the form $a\times b$ where $(\tau(a), \tau(b)) \in \{ (2,6), (3,4) \}$ and $\gcd(a,b) = 1$. You have some sort of recursion. The subproblems are much easier. – mookid Jan 7 '15 at 10:54
• Nice observation. – archangel89 Jan 7 '15 at 11:29

I know you guys love your maths (and I do too) but, for something as finite as this (i.e., not something that would consume years of compute time), you can test it with an exhaustive computer program, one that takes about three thousandths of a second on my desktop:

#include <stdio.h>

int main (int argc, char *argv[]) {
int quantN[200] = {0};

for (int number = 1; number < 200; number++) {
int quant = 0;

for (int divisor = 1; divisor <= number; divisor++)
if (number % divisor == 0)
quant++;

printf ("%3d: (%2d):", number, quant);
quantN[quant]++;

for (int divisor = 1; divisor <= number; divisor++)
if (number % divisor == 0)
printf (" %3d", divisor);

putchar ('\n');
}

putchar ('\n');
for (int quant = 0; quant < 200; quant++)
if (quantN[quant] > 0)
printf ("Quantity with %2d factors is %d\n",
quant, quantN[quant]);

return 0;
}

The relevant lines of that output are:

60: (12):   1   2   3   4   5   6  10  12  15  20  30  60
72: (12):   1   2   3   4   6   8   9  12  18  24  36  72
84: (12):   1   2   3   4   6   7  12  14  21  28  42  84
90: (12):   1   2   3   5   6   9  10  15  18  30  45  90
96: (12):   1   2   3   4   6   8  12  16  24  32  48  96
108: (12):   1   2   3   4   6   9  12  18  27  36  54 108
126: (12):   1   2   3   6   7   9  14  18  21  42  63 126
132: (12):   1   2   3   4   6  11  12  22  33  44  66 132
140: (12):   1   2   4   5   7  10  14  20  28  35  70 140
150: (12):   1   2   3   5   6  10  15  25  30  50  75 150
156: (12):   1   2   3   4   6  12  13  26  39  52  78 156
160: (12):   1   2   4   5   8  10  16  20  32  40  80 160
198: (12):   1   2   3   6   9  11  18  22  33  66  99 198

Quantity with 12 factors is 13

The complete output of that is shown below, and I'll ask in advance for forgiveness for butchering the English language with the phrase "Quantity with 1 factors is 1" but I don't see the need for making the program grammatically aware for what is, in essence, a one-off activity.

Or you can just take that to mean I'm basically lazy :-)

1: ( 1):   1
2: ( 2):   1   2
3: ( 2):   1   3
4: ( 3):   1   2   4
5: ( 2):   1   5
6: ( 4):   1   2   3   6
7: ( 2):   1   7
8: ( 4):   1   2   4   8
9: ( 3):   1   3   9
10: ( 4):   1   2   5  10
11: ( 2):   1  11
12: ( 6):   1   2   3   4   6  12
13: ( 2):   1  13
14: ( 4):   1   2   7  14
15: ( 4):   1   3   5  15
16: ( 5):   1   2   4   8  16
17: ( 2):   1  17
18: ( 6):   1   2   3   6   9  18
19: ( 2):   1  19
20: ( 6):   1   2   4   5  10  20
21: ( 4):   1   3   7  21
22: ( 4):   1   2  11  22
23: ( 2):   1  23
24: ( 8):   1   2   3   4   6   8  12  24
25: ( 3):   1   5  25
26: ( 4):   1   2  13  26
27: ( 4):   1   3   9  27
28: ( 6):   1   2   4   7  14  28
29: ( 2):   1  29
30: ( 8):   1   2   3   5   6  10  15  30
31: ( 2):   1  31
32: ( 6):   1   2   4   8  16  32
33: ( 4):   1   3  11  33
34: ( 4):   1   2  17  34
35: ( 4):   1   5   7  35
36: ( 9):   1   2   3   4   6   9  12  18  36
37: ( 2):   1  37
38: ( 4):   1   2  19  38
39: ( 4):   1   3  13  39
40: ( 8):   1   2   4   5   8  10  20  40
41: ( 2):   1  41
42: ( 8):   1   2   3   6   7  14  21  42
43: ( 2):   1  43
44: ( 6):   1   2   4  11  22  44
45: ( 6):   1   3   5   9  15  45
46: ( 4):   1   2  23  46
47: ( 2):   1  47
48: (10):   1   2   3   4   6   8  12  16  24  48
49: ( 3):   1   7  49
50: ( 6):   1   2   5  10  25  50
51: ( 4):   1   3  17  51
52: ( 6):   1   2   4  13  26  52
53: ( 2):   1  53
54: ( 8):   1   2   3   6   9  18  27  54
55: ( 4):   1   5  11  55
56: ( 8):   1   2   4   7   8  14  28  56
57: ( 4):   1   3  19  57
58: ( 4):   1   2  29  58
59: ( 2):   1  59
60: (12):   1   2   3   4   5   6  10  12  15  20  30  60
61: ( 2):   1  61
62: ( 4):   1   2  31  62
63: ( 6):   1   3   7   9  21  63
64: ( 7):   1   2   4   8  16  32  64
65: ( 4):   1   5  13  65
66: ( 8):   1   2   3   6  11  22  33  66
67: ( 2):   1  67
68: ( 6):   1   2   4  17  34  68
69: ( 4):   1   3  23  69
70: ( 8):   1   2   5   7  10  14  35  70
71: ( 2):   1  71
72: (12):   1   2   3   4   6   8   9  12  18  24  36  72
73: ( 2):   1  73
74: ( 4):   1   2  37  74
75: ( 6):   1   3   5  15  25  75
76: ( 6):   1   2   4  19  38  76
77: ( 4):   1   7  11  77
78: ( 8):   1   2   3   6  13  26  39  78
79: ( 2):   1  79
80: (10):   1   2   4   5   8  10  16  20  40  80
81: ( 5):   1   3   9  27  81
82: ( 4):   1   2  41  82
83: ( 2):   1  83
84: (12):   1   2   3   4   6   7  12  14  21  28  42  84
85: ( 4):   1   5  17  85
86: ( 4):   1   2  43  86
87: ( 4):   1   3  29  87
88: ( 8):   1   2   4   8  11  22  44  88
89: ( 2):   1  89
90: (12):   1   2   3   5   6   9  10  15  18  30  45  90
91: ( 4):   1   7  13  91
92: ( 6):   1   2   4  23  46  92
93: ( 4):   1   3  31  93
94: ( 4):   1   2  47  94
95: ( 4):   1   5  19  95
96: (12):   1   2   3   4   6   8  12  16  24  32  48  96
97: ( 2):   1  97
98: ( 6):   1   2   7  14  49  98
99: ( 6):   1   3   9  11  33  99
100: ( 9):   1   2   4   5  10  20  25  50 100
101: ( 2):   1 101
102: ( 8):   1   2   3   6  17  34  51 102
103: ( 2):   1 103
104: ( 8):   1   2   4   8  13  26  52 104
105: ( 8):   1   3   5   7  15  21  35 105
106: ( 4):   1   2  53 106
107: ( 2):   1 107
108: (12):   1   2   3   4   6   9  12  18  27  36  54 108
109: ( 2):   1 109
110: ( 8):   1   2   5  10  11  22  55 110
111: ( 4):   1   3  37 111
112: (10):   1   2   4   7   8  14  16  28  56 112
113: ( 2):   1 113
114: ( 8):   1   2   3   6  19  38  57 114
115: ( 4):   1   5  23 115
116: ( 6):   1   2   4  29  58 116
117: ( 6):   1   3   9  13  39 117
118: ( 4):   1   2  59 118
119: ( 4):   1   7  17 119
120: (16):   1   2   3   4   5   6   8  10  12  15  20  24  30  40  60 120
121: ( 3):   1  11 121
122: ( 4):   1   2  61 122
123: ( 4):   1   3  41 123
124: ( 6):   1   2   4  31  62 124
125: ( 4):   1   5  25 125
126: (12):   1   2   3   6   7   9  14  18  21  42  63 126
127: ( 2):   1 127
128: ( 8):   1   2   4   8  16  32  64 128
129: ( 4):   1   3  43 129
130: ( 8):   1   2   5  10  13  26  65 130
131: ( 2):   1 131
132: (12):   1   2   3   4   6  11  12  22  33  44  66 132
133: ( 4):   1   7  19 133
134: ( 4):   1   2  67 134
135: ( 8):   1   3   5   9  15  27  45 135
136: ( 8):   1   2   4   8  17  34  68 136
137: ( 2):   1 137
138: ( 8):   1   2   3   6  23  46  69 138
139: ( 2):   1 139
140: (12):   1   2   4   5   7  10  14  20  28  35  70 140
141: ( 4):   1   3  47 141
142: ( 4):   1   2  71 142
143: ( 4):   1  11  13 143
144: (15):   1   2   3   4   6   8   9  12  16  18  24  36  48  72 144
145: ( 4):   1   5  29 145
146: ( 4):   1   2  73 146
147: ( 6):   1   3   7  21  49 147
148: ( 6):   1   2   4  37  74 148
149: ( 2):   1 149
150: (12):   1   2   3   5   6  10  15  25  30  50  75 150
151: ( 2):   1 151
152: ( 8):   1   2   4   8  19  38  76 152
153: ( 6):   1   3   9  17  51 153
154: ( 8):   1   2   7  11  14  22  77 154
155: ( 4):   1   5  31 155
156: (12):   1   2   3   4   6  12  13  26  39  52  78 156
157: ( 2):   1 157
158: ( 4):   1   2  79 158
159: ( 4):   1   3  53 159
160: (12):   1   2   4   5   8  10  16  20  32  40  80 160
161: ( 4):   1   7  23 161
162: (10):   1   2   3   6   9  18  27  54  81 162
163: ( 2):   1 163
164: ( 6):   1   2   4  41  82 164
165: ( 8):   1   3   5  11  15  33  55 165
166: ( 4):   1   2  83 166
167: ( 2):   1 167
168: (16):   1   2   3   4   6   7   8  12  14  21  24  28  42  56  84 168
169: ( 3):   1  13 169
170: ( 8):   1   2   5  10  17  34  85 170
171: ( 6):   1   3   9  19  57 171
172: ( 6):   1   2   4  43  86 172
173: ( 2):   1 173
174: ( 8):   1   2   3   6  29  58  87 174
175: ( 6):   1   5   7  25  35 175
176: (10):   1   2   4   8  11  16  22  44  88 176
177: ( 4):   1   3  59 177
178: ( 4):   1   2  89 178
179: ( 2):   1 179
180: (18):   1   2   3   4   5   6   9  10  12  15  18  20  30  36  45  60  90 180
181: ( 2):   1 181
182: ( 8):   1   2   7  13  14  26  91 182
183: ( 4):   1   3  61 183
184: ( 8):   1   2   4   8  23  46  92 184
185: ( 4):   1   5  37 185
186: ( 8):   1   2   3   6  31  62  93 186
187: ( 4):   1  11  17 187
188: ( 6):   1   2   4  47  94 188
189: ( 8):   1   3   7   9  21  27  63 189
190: ( 8):   1   2   5  10  19  38  95 190
191: ( 2):   1 191
192: (14):   1   2   3   4   6   8  12  16  24  32  48  64  96 192
193: ( 2):   1 193
194: ( 4):   1   2  97 194
195: ( 8):   1   3   5  13  15  39  65 195
196: ( 9):   1   2   4   7  14  28  49  98 196
197: ( 2):   1 197
198: (12):   1   2   3   6   9  11  18  22  33  66  99 198
199: ( 2):   1 199

Quantity with  1 factors is 1
Quantity with  2 factors is 46
Quantity with  3 factors is 6
Quantity with  4 factors is 59
Quantity with  5 factors is 2
Quantity with  6 factors is 27
Quantity with  7 factors is 1
Quantity with  8 factors is 31
Quantity with  9 factors is 3
Quantity with 10 factors is 5
Quantity with 12 factors is 13
Quantity with 14 factors is 1
Quantity with 15 factors is 1
Quantity with 16 factors is 2
Quantity with 18 factors is 1

I think you missed two possibilities: $108=2^2\cdot 3^3$ and $198=2\cdot 3^2\cdot 11$.

• Is there any general way? – archangel89 Jan 7 '15 at 9:49
• I'm not aware of a better idea than what you did. – paw88789 Jan 7 '15 at 9:53

A general way.

Hint: Try use the function $\tau(n)$ the number of divisors of $n.$

• Do you have anything more specific in mind? We are trying to compute $\#\{n\leq 200:\tau(n)=12\}$, which is easier to write but doesn't seem to make the problem any easier. – Eric Stucky Jan 7 '15 at 10:22