72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross and also six dozen (i.e., 60 in duodecimal).
In mathematics[edit]Seventy-two is a pronic number, as it is the product of 8 and 9.[1] It is the smallest Achilles number, as it is a powerful number that is not itself a power.[2]
72 is an abundant number.[3] With exactly twelve positive divisors, including 12 (one of only two sublime numbers),[4] 72 is also the twelfth member in the sequence of refactorable numbers.[5] As no smaller number has more than 12 divisors, 72 is a largely composite number.[6] 72 has an Euler totient of 24.[7] It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72.[8] It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square of 24.[8] While 17 different integers he a totient value of 72, the sum of Euler's totient function φ(x) over the first 15 integers is 72.[9] It is also a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).[10]
72 is the second multiple of 12, after 48, that is not a sum of twin primes. It is, however, the sum of four consecutive primes (13 + 17 + 19 + 23),[11] as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).[12]72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: 112 − 72 = 192 − 172.[13]72 is the sum of the first two sphenic numbers (30, 42),[14] which he a difference of 12, that is also their abundance.[15][16]72 is the magic constant of the first non-normal, full prime reciprocal magic square in decimal, based on 1/17 in a 16 × 16 grid.[17][18]72 is the sum between 60 and 12, the former being the second unitary perfect number before 6 (and the latter the smallest of only two sublime numbers). More specifically, twelve is also the number of divisors of 60, as the smallest number with this many divisors.[19]72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30.[20]72 is the sum of the eighth row of Lozanić's triangle, and equal to the sum of the previous four rows (36, 20, 10, 6).[21] As such, this row is the third and largest to be in equivalence with a sum of consecutive k row sums, after (1, 2, 3; 6) and (6, 10, 20; 36).72 is the number of degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge.72 plays a role in the Rule of 72 in economics when approximating annual compounding of interest rates of a round 6% to 10%, due in part to its high number of divisors.
Inside E n {\displaystyle \mathrm {E} _{n}} Lie algebras:
72 is the number of vertices of the six-dimensional 122 polytope, which also contains as facets 720 edges, 702 polychoral 4-faces, of which 270 are four-dimensional 16-cells, and two sets of 27 demipenteract 5-faces. These 72 vertices are the root vectors of the simple Lie group E 6 {\displaystyle \mathrm {E} _{6}} , which as a honeycomb under 222 forms the E 6 {\displaystyle \mathrm {E} _{6}} lattice. 122 is part of a family of k22 polytopes whose first member is the fourth-dimensional 3-3 duoprism, of symmetry order 72 and made of six triangular prisms. On the other hand, 321 ∈ k21 is the only semiregular polytope in the seventh dimension, also featuring a total of 702 6-faces of which 576 are 6-simplexes and 126 are 6-orthoplexes that contain 60 edges and 12 vertices, or collectively 72 one-dimensional and two-dimensional elements; with 126 the number of root vectors in E 7 {\displaystyle \mathrm {E} _{7}} , which are contained in the vertices of 231 ∈ k31, also with 576 or 242 6-simplexes like 321. The triangular prism is the root polytope in the k21 family of polytopes, which is the simplest semiregular polytope, with k31 rooted in the analogous four-dimensional tetrahedral prism that has four triangular prisms alongside two tetrahedra as cells. The complex Hessian polyhedron in C 3 {\displaystyle \mathbb {C} ^{3}} contains 72 regular complex triangular edges, as well as 27 polygonal Möbius–Kantor faces and 27 vertices. It is notable for being the vertex figure of the complex Witting polytope, which shares 240 vertices with the eight-dimensional semiregular 421 polytope whose vertices in turn represent the root vectors of the simple Lie group E 8 {\displaystyle \mathrm {E} _{8}} .
Lie algebras:
, which as a honeycomb under 222 forms the
E
6
{\displaystyle \mathrm {E} _{6}}
, which are contained in the vertices of 231 ∈ k31, also with 576 or 242 6-simplexes like 321. The triangular prism is the root polytope in the k21 family of polytopes, which is the simplest semiregular polytope, with k31 rooted in the analogous four-dimensional tetrahedral prism that has four triangular prisms alongside two tetrahedra as cells.
The complex Hessian polyhedron in
C
3
{\displaystyle \mathbb {C} ^{3}}
contains 72 regular complex triangular edges, as well as 27 polygonal Möbius–Kantor faces and 27 vertices. It is notable for being the vertex figure of the complex Witting polytope, which shares 240 vertices with the eight-dimensional semiregular 421 polytope whose vertices in turn represent the root vectors of the simple Lie group
E
8
{\displaystyle \mathrm {E} _{8}}
.
There are 72 compact and paracompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in only three-dimensional and four-dimensional spaces, with the remaining 58 paracompact or noncompact infinite representations in dimensions three through nine. These terminate with three paracompact groups in the ninth dimension, of which the most important is T ~ 9 {\displaystyle {\tilde {T}}_{9}} : it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its vertex figure, which is the geometric representation of the E 8 {\displaystyle \mathrm {E} _{8}} lattice. Furthermore, T ~ 9 {\displaystyle {\tilde {T}}_{9}} shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form E 8 {\displaystyle \mathrm {E} _{8}} ++ equivalent to the tenth-dimensional symmetries of Lie algebra E 10 {\displaystyle \mathrm {E} _{10}} .
: it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its vertex figure, which is the geometric representation of the
E
8
{\displaystyle \mathrm {E} _{8}}
.
72 lies between the 8th pair of twin primes (71, 73), where 71 is the largest supersingular prime that is a factor of the largest sporadic group (the friendly giant F 1 {\displaystyle \mathbb {F_{1}} } ), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers[23][a] that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of F 1 {\displaystyle \mathbb {F_{1}} } .[24] Sporadic groups are a family of twenty-six finite simple groups, where E 6 {\displaystyle \mathrm {E} _{6}} , E 7 {\displaystyle \mathrm {E} _{7}} , and E 8 {\displaystyle \mathrm {E} _{8}} are associated exceptional groups that are part of sixteen finite Lie groups that are also simple, or non-trivial groups whose only normal subgroups are the trivial group and the groups themselves.[b]
In religion[edit] In Islam, 72 is the number of beautiful wives that are promised to martyrs in paradise, according to Hadith (sayings of Muhammad).[25][26][relevant?] In other fields[edit]
), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers[23][a] that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of
F
1
{\displaystyle \mathbb {F_{1}} }
Seventy-two is also:
In typography, a point is 1/72 inch.[27] The Rule of 72 in finance. 72 equal temperament is a tuning used in Byzantine music and by some modern composers. The number of micro seasons in the traditional Japanese calendar[28] Notes[edit] ^ Where 71 is also the largest prime number less than 73 that is not a member of this set. ^ The only other finite simple groups are the infinite families of cyclic groups and alternating groups. An exception is the Tits group T {\displaystyle \mathbb {T} } , which is sometimes considered a 17th non-strict group of Lie type that can otherwise more loosely classify as a 27th sporadic group. References[edit] ^ Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15. ^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15. The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ... ^ Sloane, N. J. A. (ed.). "Sequence A067128 (Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ a b Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02. ^ Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02. ^ Sloane, N. J. A. (ed.). "Sequence A090788 (Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03. ^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13. ^ Sloane, N. J. A. (ed.). "Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13. ^ Subramani, K. (2020). "On two interesting properties of primes, p, with reciprocals in base 10 hing maximum period p - 1" (PDF). J. Of Math. Sci. & Comp. Math. 1 (2). Auburn, WA: S.M.A.R.T.: 198–200. doi:10.15864/jmscm.1204 (inactive 1 July 2025). eISSN 2644-3368. S2CID 235037714.{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link) ^ Sloane, N. J. A. (ed.). "Sequence A007450 (Decimal expansion of 1/17.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-24. ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-11. ^ Sloane, N. J. A. (ed.). "Sequence A200720 (Number of distinct normal magic stars of type {n/2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-09. ^ Sloane, N. J. A. (ed.). "Sequence A005418 (...row sums of Losanitsch's triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22. ^ Did Wells: The Penguin Dictionary of Curious and Interesting Numbers ^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bharga's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73} ^ He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv:1505.06742 [math.AG]. ^ Jami`at-Tirmidhi. "The Book on Virtues of Jihad, Vol. 3, Book 20, Hadith 1663". Sunnah.com - Sayings and Teachings of Prophet Muhammad (صلى الله عليه و سلم). Retrieved 2024-04-02. ^ Kruglanski, Arie W.; Chen, Xiaoyan; Dechesne, Mark; Fishman, Shira; Orehek, Edward (2009). "Fully Committed: Suicide Bombers' Motivation and the Quest for Personal Significance". Political Psychology. 30 (3): 331–357. doi:10.1111/j.1467-9221.2009.00698.x. ISSN 0162-895X. JSTOR 25655398. ^ W3C. "CSS Units". w3.org. Retrieved September 28, 2024.{{cite web}}: CS1 maint: numeric names: authors list (link) ^ "Japan's 72 Microseasons". 16 October 2015. External links[edit] Go Figure: What can 72 tell us about life, BBC News, 20 July 2011 vte Z {\displaystyle \mathbb {Z} } Integers−2, −10 to 1990 to 99100 to 199 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 to 399200 to 299300 to 399 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 to 999400s, 500s, and 600s700s, 800s, and 900s 400 420 440 495 496 500 501 511 512 555 600 610 613 616 666 693 700 720 743 744 777 786 800 801 836 840 880 881 888 900 911 971 987 999 1000s and 10,000s1000s 1000 1001 1023 1024 1089 1093 1105 1234 1289 1458 1510 1728 1729 1980 1987 2000 2016 2520 3000 3511 4000 4104 5000 5040 6000 6174 7000 7744 7825 8000 8128 8192 9000 9855 9999 10,000s 10,000 16,807 20,000 30,000 40,000 50,000 60,000 64,079 65,535 65,536 65,537 70,000 80,000 90,000 100,000s to 10,000,000,000,000s 100,000 142,857 144,000 1,000,000 10,000,000 43,112,609 100,000,000 1,000,000,000 2,147,483,647 4,294,967,295 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000 Large numbers
, which is sometimes considered a 17th non-strict group of Lie type that can otherwise more loosely classify as a 27th sporadic group.
References[edit]
^ Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ...
^ Sloane, N. J. A. (ed.). "Sequence A067128 (Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ a b Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02.
^ Sloane, N. J. A. (ed.). "Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02.
^ Sloane, N. J. A. (ed.). "Sequence A090788 (Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-03.
^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
^ Sloane, N. J. A. (ed.). "Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-13.
^ Subramani, K. (2020). "On two interesting properties of primes, p, with reciprocals in base 10 hing maximum period p - 1" (PDF). J. Of Math. Sci. & Comp. Math. 1 (2). Auburn, WA: S.M.A.R.T.: 198–200. doi:10.15864/jmscm.1204 (inactive 1 July 2025). eISSN 2644-3368. S2CID 235037714.{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link)
^ Sloane, N. J. A. (ed.). "Sequence A007450 (Decimal expansion of 1/17.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-24.
^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-11.
^ Sloane, N. J. A. (ed.). "Sequence A200720 (Number of distinct normal magic stars of type {n/2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-09.
^ Sloane, N. J. A. (ed.). "Sequence A005418 (...row sums of Losanitsch's triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
^ Did Wells: The Penguin Dictionary of Curious and Interesting Numbers
^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bharga's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}
^ He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv:1505.06742 [math.AG].
^ Jami`at-Tirmidhi. "The Book on Virtues of Jihad, Vol. 3, Book 20, Hadith 1663". Sunnah.com - Sayings and Teachings of Prophet Muhammad (صلى الله عليه و سلم). Retrieved 2024-04-02.
^ Kruglanski, Arie W.; Chen, Xiaoyan; Dechesne, Mark; Fishman, Shira; Orehek, Edward (2009). "Fully Committed: Suicide Bombers' Motivation and the Quest for Personal Significance". Political Psychology. 30 (3): 331–357. doi:10.1111/j.1467-9221.2009.00698.x. ISSN 0162-895X. JSTOR 25655398.
^ W3C. "CSS Units". w3.org. Retrieved September 28, 2024.{{cite web}}: CS1 maint: numeric names: authors list (link)
^ "Japan's 72 Microseasons". 16 October 2015.
External links[edit]
Go Figure: What can 72 tell us about life, BBC News, 20 July 2011
vte
Z
{\displaystyle \mathbb {Z} }
Integers−2, −10 to 1990 to 99100 to 199
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200 to 399200 to 299300 to 399
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400 to 999400s, 500s, and 600s700s, 800s, and 900s
400
420
440
495
496
500
501
511
512
555
600
610
613
616
666
693
700
720
743
744
777
786
800
801
836
840
880
881
888
900
911
971
987
999
1000s and 10,000s1000s
1000
1001
1023
1024
1089
1093
1105
1234
1289
1458
1510
1728
1729
1980
1987
2000
2016
2520
3000
3511
4000
4104
5000
5040
6000
6174
7000
7744
7825
8000
8128
8192
9000
9855
9999
10,000s
10,000
16,807
20,000
30,000
40,000
50,000
60,000
64,079
65,535
65,536
65,537
70,000
80,000
90,000
100,000s to 10,000,000,000,000s
100,000
142,857
144,000
1,000,000
10,000,000
43,112,609
100,000,000
1,000,000,000
2,147,483,647
4,294,967,295
10,000,000,000
100,000,000,000
1,000,000,000,000
10,000,000,000,000
Large numbers