List of 310 solved problems

The difficulty rating is two-fold: the percentage refers to the Project Euler rating,
while the easy/medium/hard/advanced classification was made by Hackerrank where the problem is often modified (wording and/or input range).

problem difficulty solved live test
1 Multiples of 3 and 5 5% easy February 22, 2017 yes
2 Even Fibonacci numbers 5% easy February 23, 2017 yes
3 Largest prime factor 5% easy February 23, 2017 yes
4 Largest palindrome product 5% medium February 23, 2017 yes
5 Smallest multiple 5% medium February 23, 2017 yes
6 Sum square difference 5% easy February 23, 2017 yes
7 10001st prime 5% easy February 23, 2017 yes
8 Largest product in a series 5% easy February 23, 2017 yes
9 Special Pythagorean triplet 5% easy February 23, 2017 yes
10 Summation of primes 5% medium February 23, 2017 yes
11 Largest product in a grid 5% easy February 23, 2017 yes
12 Highly divisible triangular number 5% easy February 23, 2017 yes
13 Large sum 5% easy February 23, 2017 yes
14 Longest Collatz sequence 5% easy February 23, 2017 yes
15 Lattice paths 5% easy February 23, 2017 yes
16 Power digit sum 5% easy February 23, 2017 yes
17 Number letter counts 5% easy February 24, 2017 yes
18 Maximum path sum I 5% easy February 24, 2017 yes
19 Counting Sundays 5% easy February 24, 2017 yes
20 Factorial digit sum 5% easy February 24, 2017 yes
21 Amicable numbers 5% easy February 24, 2017 yes
22 Names scores 5% easy February 23, 2017 no
23 Non-abundant sums 5% easy February 24, 2017 yes
24 Lexicographic permutations 5% easy February 23, 2017 yes
25 1000-digit Fibonacci number 5% easy February 24, 2017 yes
26 Reciprocal cycles 5% easy February 24, 2017 yes
27 Quadratic primes 5% easy February 23, 2017 yes
28 Number spiral diagonals 5% easy February 23, 2017 yes
29 Distinct powers 5% easy February 24, 2017 yes
30 Digit fifth powers 5% easy February 23, 2017 yes
31 Coin sums 5% easy February 23, 2017 yes
32 Pandigital products 5% easy February 23, 2017 yes
33 Digit cancelling fractions 5% hard February 24, 2017 yes
34 Digit factorials 5% easy February 23, 2017 yes
35 Circular primes 5% easy February 24, 2017 yes
36 Double-base palindromes 5% easy February 24, 2017 yes
37 Truncatable primes 5% easy February 23, 2017 yes
38 Pandigital multiples 5% easy February 25, 2017 yes
39 Integer right triangles 5% easy February 25, 2017 yes
40 Champernowne's constant 5% medium February 25, 2017 yes
41 Pandigital prime 5% easy February 25, 2017 yes
42 Coded triangle numbers 5% easy February 27, 2017 yes
43 Sub-string divisibility 5% easy February 23, 2017 yes
44 Pentagon numbers 5% easy February 27, 2017 yes
45 Triangular, pentagonal, and hexagonal 5% easy February 27, 2017 yes
46 Goldbach's other conjecture 5% easy February 27, 2017 yes
47 Distinct primes factors 5% easy February 26, 2017 yes
48 Self powers 5% easy February 26, 2017 yes
49 Prime permutations 5% hard February 27, 2017 yes
50 Consecutive prime sum 5% hard February 27, 2017 yes
51 Prime digit replacements 15% advanced February 27, 2017 yes
52 Permuted multiples 5% easy February 27, 2017 yes
53 Combinatoric selections 5% easy February 27, 2017 yes
54 Poker hands 10% easy February 28, 2017 yes
55 Lychrel numbers 5% easy February 27, 2017 yes
56 Powerful digit sum 5% easy February 28, 2017 yes
57 Square root convergents 5% easy February 28, 2017 yes
58 Spiral primes 5% easy February 28, 2017 yes
59 XOR decryption 5% easy March 1, 2017 yes
60 Prime pair sets 20% hard March 1, 2017 yes
61 Cyclical figurate numbers 20% easy March 6, 2017 yes
62 Cubic permutations 15% easy March 1, 2017 yes
63 Powerful digit counts 5% easy February 28, 2017 yes
64 Odd period square roots 20% easy March 8, 2017 yes
65 Convergents of e 15% easy March 8, 2017 yes
66 Diophantine equation 25% hard March 12, 2017 yes
67 Maximum path sum II 5% medium February 24, 2017 yes
68 Magic 5-gon ring 25% hard March 13, 2017 yes
69 Totient maximum 10% easy March 1, 2017 yes
70 Totient permutation 20% easy March 11, 2017 yes
71 Ordered fractions 10% easy March 13, 2017 yes
72 Counting fractions 20% easy March 13, 2017 yes
73 Counting fractions in a range 15% easy March 15, 2017 yes
74 Digit factorial chains 15% easy March 12, 2017 yes
75 Singular integer right triangles 25% easy March 12, 2017 yes
76 Counting summations 10% easy March 13, 2017 yes
77 Prime summations 25% easy March 17, 2017 yes
78 Coin partitions 30% medium March 16, 2017 yes
79 Passcode derivation 5% easy March 13, 2017 yes
80 Square root digital expansion 20% easy March 14, 2017 yes
81 Path sum: two ways 10% easy March 12, 2017 yes
82 Path sum: three ways 20% easy March 12, 2017 yes
83 Path sum: four ways 25% hard March 12, 2017 yes
84 Monopoly odds 35% hard March 17, 2017 yes
85 Counting rectangles 15% medium March 12, 2017 yes
86 Cuboid route 35% hard March 15, 2017 yes
87 Prime power triples 20% easy March 12, 2017 yes
88 Product-sum numbers 40% easy March 19, 2017 yes
89 Roman numerals 20% easy March 15, 2017 yes
90 Cube digit pairs 40% easy March 19, 2017 yes
91 Right triangles with integer coordinates 25% medium March 15, 2017 yes
92 Square digit chains 5% easy March 9, 2017 yes
93 Arithmetic expressions 15% medium March 16, 2017 yes
94 Almost equilateral triangles 35% medium March 16, 2017 yes
95 Amicable chains 30% medium March 16, 2017 yes
96 Su Doku 25% hard March 13, 2017 yes
97 Large non-Mersenne prime 5% hard March 2, 2017 yes
98 Anagramic squares 35% hard March 14, 2017 yes
99 Largest exponential 10% medium March 1, 2017 yes
100 Arranged probability 30% advanced March 15, 2017 yes
101 Optimum polynomial 35% May 23, 2017 yes
102 Triangle containment 15% easy May 11, 2017 yes
103 Special subset sums: optimum 45% easy May 16, 2017 yes
104 Pandigital Fibonacci ends 25% easy May 12, 2017 yes
105 Special subset sums: testing 45% easy May 16, 2017 yes
106 Special subset sums: meta-testing 50% August 8, 2017 yes
107 Minimal network 35% easy May 11, 2017 no
108 Diophantine reciprocals I 30% easy May 12, 2017 yes
109 Darts 45% May 17, 2017 yes
110 Diophantine reciprocals II 40% easy July 28, 2017 yes
111 Primes with runs 45% easy May 17, 2017 yes
112 Bouncy numbers 15% advanced May 11, 2017 yes
113 Non-bouncy numbers 30% easy May 15, 2017 yes
114 Counting block combinations I 35% medium May 13, 2017 yes
115 Counting block combinations II 35% medium May 13, 2017 yes
116 Red, green or blue tiles 30% medium May 14, 2017 yes
117 Red, green, and blue tiles 35% easy May 18, 2017 yes
118 Pandigital prime sets 45% hard May 18, 2017 yes
119 Digit power sum 30% easy May 13, 2017 yes
120 Square remainders 25% easy May 11, 2017 no
121 Disc game prize fund 35% easy May 19, 2017 yes
122 Efficient exponentiation 40% easy May 15, 2017 yes
123 Prime square remainders 30% May 11, 2017 no
124 Ordered radicals 25% advanced May 19, 2017 yes
125 Palindromic sums 25% easy May 17, 2017 yes
126 Cuboid layers 55% August 18, 2017 yes
127 abc-hits 50% June 26, 2017 yes
128 Hexagonal tile differences 55% hard June 24, 2017 yes
129 Repunit divisibility 45% hard May 21, 2017 yes
130 Composites with prime repunit property 45% easy July 17, 2017 yes
131 Prime cube partnership 40% hard June 27, 2017 yes
132 Large repunit factors 45% June 1, 2017 yes
133 Repunit nonfactors 50% medium June 2, 2017 yes
134 Prime pair connection 45% May 24, 2017 no
135 Same differences 45% medium May 19, 2017 yes
136 Singleton differences 45% May 19, 2017 no
137 Fibonacci golden nuggets 50% easy August 1, 2017 yes
138 Special isosceles triangles 45% easy July 10, 2017 yes
139 Pythagorean tiles 50% June 1, 2017 yes
141 Investigating progressive numbers, n, ... 60% September 1, 2017 yes
142 Perfect Square Collection 45% June 27, 2017 no
144 Investigating multiple reflections of ... 50% June 30, 2017 no
145 How many reversible numbers are there ... 20% May 23, 2017 no
146 Investigating a Prime Pattern 50% easy June 16, 2017 yes
147 Rectangles in cross-hatched grids 65% hard August 9, 2017 yes
148 Exploring Pascal's triangle 50% July 12, 2017 yes
149 Searching for a maximum-sum subsequence 50% May 25, 2017 no
150 Searching a triangular array for a sub ... 55% easy July 5, 2017 yes
151 Paper sheets of standard sizes: an expected ... 35% May 23, 2017 no
152 Writing 1/2 as a sum of inverse squares 65% advanced September 17, 2017 yes
154 Exploring Pascal's pyramid 60% hard July 13, 2017 yes
155 Counting Capacitor Circuits 60% easy September 15, 2017 yes
156 Counting Digits 70% medium October 6, 2017 yes
158 Exploring strings for which only one character ... 55% June 13, 2017 yes
159 Digital root sums of factorisations 60% medium June 22, 2017 no
160 Factorial trailing digits 60% July 13, 2017 yes
161 Triominoes 70% hard June 29, 2017 yes
162 Hexadecimal numbers 45% medium May 25, 2017 yes
163 Cross-hatched triangles 70% easy July 7, 2017 yes
164 Numbers for which no three consecutive ... 45% easy June 16, 2017 yes
165 Intersections 65% easy June 20, 2017 yes
166 Criss Cross 50% easy June 14, 2017 yes
168 Number Rotations 65% easy August 3, 2017 yes
169 Exploring the number of different ways ... 50% easy May 30, 2017 yes
170 Find the largest 0 to 9 pandigital that ... 70% easy June 19, 2017 yes
171 Finding numbers for which the sum of the ... 70% medium July 7, 2017 yes
172 Investigating numbers with few repeated ... 55% June 15, 2017 yes
173 Using up to one million tiles how many ... 30% medium May 22, 2017 yes
174 Counting the number of "hollow" square ... 40% easy May 23, 2017 yes
178 Step Numbers 55% medium June 28, 2017 yes
179 Consecutive positive divisors 25% easy May 16, 2017 yes
181 Investigating in how many ways objects ... 70% easy June 28, 2017 yes
182 RSA encryption 60% September 3, 2017 yes
183 Maximum product of parts 45% easy June 13, 2017 yes
185 Number Mind 55% medium May 30, 2017 yes
186 Connectedness of a network 60% easy July 4, 2017 yes
187 Semiprimes 20% medium May 23, 2017 yes
188 The hyperexponentiation of a number 35% June 14, 2017 yes
189 Tri-colouring a triangular grid 70% easy September 5, 2017 yes
190 Maximising a weighted product 50% June 27, 2017 yes
191 Prize Strings 45% May 25, 2017 yes
193 Squarefree Numbers 55% June 15, 2017 yes
196 Prime triplets 65% August 19, 2017 yes
197 Investigating the behaviour of a recursively ... 45% June 13, 2017 no
199 Iterative Circle Packing 70% September 10, 2017 yes
200 Find the 200th prime-proof sqube containing ... 65% August 26, 2017 yes
201 Subsets with a unique sum 65% July 2, 2017 yes
203 Squarefree Binomial Coefficients 25% May 31, 2017 yes
204 Generalised Hamming Numbers 30% May 27, 2017 yes
205 Dice Game 15% May 26, 2017 yes
206 Concealed Square 5% May 26, 2017 no
207 Integer partition equations 40% May 31, 2017 yes
208 Robot Walks 70% November 11, 2017 yes
209 Circular Logic 60% September 12, 2017 no
211 Divisor Square Sum 50% May 31, 2017 yes
213 Flea Circus 60% July 5, 2017 yes
214 Totient Chains 40% June 20, 2017 yes
215 Crack-free Walls 50% July 22, 2017 yes
216 Investigating the primality of numbers ... 45% November 1, 2017 yes
218 Perfect right-angled triangles 55% August 28, 2017 no
219 Skew-cost coding 70% July 30, 2017 yes
222 Sphere Packing 60% July 24, 2017 yes
225 Tribonacci non-divisors 45% June 23, 2017 yes
226 A Scoop of Blancmange 65% September 18, 2017 yes
227 The Chase 65% August 24, 2017 yes
229 Four Representations using Squares 70% June 21, 2017 yes
230 Fibonacci Words 50% July 12, 2017 yes
231 The prime factorisation of binomial coefficients 40% June 28, 2017 yes
232 The Race 65% July 21, 2017 yes
234 Semidivisible numbers 50% August 5, 2017 yes
235 An Arithmetic Geometric sequence 40% July 2, 2017 no
237 Tours on a 4 x n playing board 70% October 17, 2017 yes
239 Twenty-two Foolish Primes 65% September 9, 2017 yes
240 Top Dice 60% July 20, 2017 yes
243 Resilience 35% June 19, 2017 no
244 Sliders 70% September 25, 2017 yes
247 Squares under a hyperbola 65% August 14, 2017 yes
248 Numbers for which Euler's totient function ... 70% November 12, 2017 yes
249 Prime Subset Sums 60% July 14, 2017 yes
250 250250 55% July 14, 2017 yes
259 Reachable Numbers 70% July 4, 2017 yes
260 Stone Game 70% August 4, 2017 yes
265 Binary Circles 40% July 1, 2017 yes
266 Pseudo Square Root 65% July 3, 2017 yes
267 Billionaire 50% July 11, 2017 yes
268 Counting numbers with at least four distinct ... 70% September 27, 2017 yes
273 Sum of Squares 70% September 14, 2017 yes
274 Divisibility Multipliers 65% August 23, 2017 yes
277 A Modified Collatz sequence 35% July 11, 2017 yes
278 Linear Combinations of Semiprimes 50% July 11, 2017 yes
279 Triangles with integral sides and an integral ... 60% December 27, 2017 yes
280 Ant and seeds 65% August 25, 2017 yes
284 Steady Squares 55% October 13, 2017 yes
286 Scoring probabilities 50% July 6, 2017 yes
287 Quadtree encoding (a simple compression ... 40% August 29, 2017 yes
288 An enormous factorial 35% October 19, 2017 yes
290 Digital Signature 50% November 14, 2017 yes
291 Panaitopol Primes 45% July 7, 2017 yes
293 Pseudo-Fortunate Numbers 25% July 26, 2017 yes
297 Zeckendorf Representation 35% July 25, 2017 yes
300 Protein folding 50% August 27, 2017 yes
301 Nim 15% July 4, 2017 yes
303 Multiples with small digits 35% July 3, 2017 yes
304 Primonacci 35% August 7, 2017 yes
306 Paper-strip Game 55% September 16, 2017 yes
307 Chip Defects 35% August 12, 2017 yes
308 An amazing Prime-generating Automaton 65% December 11, 2017 yes
309 Integer Ladders 50% July 27, 2017 yes
310 Nim Square 40% December 12, 2017 yes
313 Sliding game 30% September 19, 2017 yes
315 Digital root clocks 20% July 29, 2017 yes
317 Firecracker 35% July 5, 2017 yes
321 Swapping Counters 30% August 30, 2017 yes
323 Bitwise-OR operations on random integers 20% July 31, 2017 yes
324 Building a tower 50% December 20, 2017 yes
327 Rooms of Doom 40% October 27, 2017 yes
329 Prime Frog 25% August 13, 2017 yes
333 Special partitions 35% September 20, 2017 yes
336 Maximix Arrangements 25% July 15, 2017 yes
341 Golomb's self-describing sequence 45% August 15, 2017 yes
343 Fractional Sequences 35% September 29, 2017 yes
345 Matrix Sum 15% July 18, 2017 no
346 Strong Repunits 15% July 9, 2017 yes
347 Largest integer divisible by two primes 15% July 17, 2017 yes
348 Sum of a square and a cube 25% July 23, 2017 yes
349 Langton's ant 35% July 19, 2017 yes
351 Hexagonal orchards 25% September 26, 2017 yes
357 Prime generating integers 10% July 8, 2017 yes
358 Cyclic numbers 25% October 26, 2017 no
359 Hilbert's New Hotel 25% September 6, 2017 yes
363 Bézier Curves 35% September 7, 2017 no
371 Licence plates 30% September 21, 2017 yes
375 Minimum of subsequences 40% December 6, 2017 yes
381 (prime-k) factorial 15% August 20, 2017 yes
386 Maximum length of an antichain 40% November 3, 2017 yes
387 Harshad Numbers 10% August 2, 2017 yes
393 Migrating ants 50% September 23, 2017 yes
401 Sum of squares of divisors 25% August 22, 2017 yes
407 Idempotents 20% August 31, 2017 yes
411 Uphill paths 45% October 5, 2017 yes
418 Factorisation triples 40% October 4, 2017 yes
425 Prime connection 25% August 6, 2017 yes
429 Sum of squares of unitary divisors 20% September 13, 2017 yes
455 Powers With Trailing Digits 40% October 1, 2017 yes
458 Permutations of Project 30% December 18, 2017 yes
461 Almost Pi 30% September 28, 2017 yes
473 Phigital number base 35% September 30, 2017 yes
485 Maximum number of divisors 30% December 8, 2017 yes
491 Double pandigital number divisible by 11 20% August 16, 2017 yes
493 Under The Rainbow 15% August 21, 2017 yes
500 Problem 500!!! 15% September 11, 2017 yes
501 Eight Divisors 40% August 17, 2017 yes
504 Square on the Inside 15% August 11, 2017 yes
510 Tangent Circles 30% September 8, 2017 yes
516 5-smooth totients 20% November 15, 2017 yes
518 Prime triples and geometric sequences 20% September 2, 2017 yes
523 First Sort I 60% September 4, 2017 yes
549 Divisibility of factorials 10% August 10, 2017 yes
563 Robot Welders 45% October 11, 2017 yes
565 Divisibility of sum of divisors 35% December 10, 2017 yes
571 Super Pandigital Numbers 25% September 22, 2017 yes
577 Counting hexagons 20% December 31, 2017 yes
581 47-smooth triangular numbers 25% October 12, 2017 yes
587 Concave triangle 20% October 23, 2017 yes
601 Divisibility streaks 20% October 10, 2017 yes
607 Marsh Crossing October 18, 2017 no
610 Roman Numerals II October 9, 2017 no
611 Hallway of square steps October 10, 2017 yes
613 Pythagorean Ant November 21, 2017 no
615 The millionth number with at least one ... December 4, 2017 no

List of 4 solved problem(s) which are not published yet

I haven't had the time yet to write a proper explanation of my solutions to these problems:
251, 355, 433, 469.

They will be published soon.

List of 4 unsolved problem(s)

I can reproduce the example but can't quite compute the correct result for these problems:

problem difficulty
412 Gnomon numbering 50%
436 Unfair wager 55%
539 Odd elimination 30%
612 Friend numbers

Heatmap

Please click on a problem's number to open my solution to that problem:

green   solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too
yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily)
gray problems are already solved but I haven't published my solution yet
blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much
orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte
red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too
black problems are solved but access to the solution is blocked for a few days until the next problem is published
[new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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
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
401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425
426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475
476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525
526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550
551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575
576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625
626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650
651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675
676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700
701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725
726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750
751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775
776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800
801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816
The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
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