1 - Multiples of 3 and 5
2 - Even Fibonacci numbers
3 - Largest prime factor
4 - Largest palindrome product
5 - Smallest multiple
6 - Sum square difference
7 - 10001st prime
8 - Largest product in a series
9 - Special Pythagorean triplet
10 - Summation of primes
11 - Largest product in a grid
12 - Highly divisible triangular number
13 - Large sum
14 - Longest Collatz sequence
15 - Lattice paths
16 - Power digit sum
17 - Number letter counts
18 - Maximum path sum I
19 - Counting Sundays
20 - Factorial digit sum
21 - Amicable numbers
22 - Names scores
23 - Non-abundant sums
24 - Lexicographic permutations
25 - 1000-digit Fibonacci number
26 - Reciprocal cycles
27 - Quadratic primes
28 - Number spiral diagonals
29 - Distinct powers
30 - Digit fifth powers
31 - Coin sums
32 - Pandigital products
33 - Digit cancelling fractions
34 - Digit factorials
35 - Circular primes
36 - Double-base palindromes
37 - Truncatable primes
38 - Pandigital multiples
39 - Integer right triangles
40 - Champernowne's constant
41 - Pandigital prime
42 - Coded triangle numbers
43 - Sub-string divisibility
44 - Pentagon numbers
45 - Triangular, pentagonal, and hexagonal
46 - Goldbach's other conjecture
47 - Distinct primes factors
48 - Self powers
49 - Prime permutations
50 - Consecutive prime sum
51 - Prime digit replacements
52 - Permuted multiples
53 - Combinatoric selections
54 - Poker hands
55 - Lychrel numbers
56 - Powerful digit sum
57 - Square root convergents
58 - Spiral primes
59 - XOR decryption
60 - Prime pair sets
61 - Cyclical figurate numbers
62 - Cubic permutations
63 - Powerful digit counts
64 - Odd period square roots
65 - Convergents of e
66 - Diophantine equation
67 - Maximum path sum II
68 - Magic 5-gon ring
69 - Totient maximum
70 - Totient permutation
71 - Ordered fractions
72 - Counting fractions
73 - Counting fractions in a range
74 - Digit factorial chains
75 - Singular integer right triangles
76 - Counting summations
77 - Prime summations
78 - Coin partitions
79 - Passcode derivation
80 - Square root digital expansion
81 - Path sum: two ways
82 - Path sum: three ways
83 - Path sum: four ways
84 - Monopoly odds
85 - Counting rectangles
86 - Cuboid route
87 - Prime power triples
88 - Product-sum numbers
89 - Roman numerals
90 - Cube digit pairs
91 - Right triangles with integer coordinates
92 - Square digit chains
93 - Arithmetic expressions
94 - Almost equilateral triangles
95 - Amicable chains
96 - Su Doku
97 - Large non-Mersenne prime
98 - Anagramic squares
99 - Largest exponential
100 - Arranged probability
101 - Optimum polynomial
102 - Triangle containment
103 - Special subset sums: optimum
104 - Pandigital Fibonacci ends
105 - Special subset sums: testing
106 - Special subset sums: meta-testing
107 - Minimal network
108 - Diophantine reciprocals I
109 - Darts
110 - Diophantine reciprocals II
111 - Primes with runs
112 - Bouncy numbers
113 - Non-bouncy numbers
114 - Counting block combinations I
115 - Counting block combinations II
116 - Red, green or blue tiles
117 - Red, green, and blue tiles
118 - Pandigital prime sets
119 - Digit power sum
120 - Square remainders
121 - Disc game prize fund
122 - Efficient exponentiation
123 - Prime square remainders
124 - Ordered radicals
125 - Palindromic sums
126 - Cuboid layers
127 - abc-hits
128 - Hexagonal tile differences
129 - Repunit divisibility
130 - Composites with prime repunit property
131 - Prime cube partnership
132 - Large repunit factors
133 - Repunit nonfactors
134 - Prime pair connection
135 - Same differences
136 - Singleton difference
137 - Fibonacci golden nuggets
138 - Special isosceles triangles
139 - Pythagorean tiles
140 - Modified Fibonacci golden nuggets
141 - Investigating progressive numbers, n, which are also square
142 - Perfect Square Collection
143 - Investigating the Torricelli point of a triangle
144 - Investigating multiple reflections of a laser beam
145 - How many reversible numbers are there below one-billion?
146 - Investigating a Prime Pattern
147 - Rectangles in cross-hatched grids
148 - Exploring Pascal's triangle
149 - Searching for a maximum-sum subsequence
150 - Searching a triangular array for a sub-triangle having minimum-sum
151 - Paper sheets of standard sizes: an expected-value problem
152 - Writing 1/2 as a sum of inverse squares
153 - Investigating Gaussian Integers
154 - Exploring Pascal's pyramid
155 - Counting Capacitor Circuits
156 - Counting Digits
157 - Solving the diophantine equation 1/a+1/b= p/10n
158 - Exploring strings for which only one character comes lexicographically after its neighbour to the left
159 - Digital root sums of factorisations
160 - Factorial trailing digits
161 - Triominoes
162 - Hexadecimal numbers
163 - Cross-hatched triangles
164 - Numbers for which no three consecutive digits have a sum greater than a given value
165 - Intersections
166 - Criss Cross
167 - Investigating Ulam sequences
168 - Number Rotations
169 - Exploring the number of different ways a number can be expressed as a sum of powers of 2
170 - Find the largest 0 to 9 pandigital that can be formed by concatenating products
171 - Finding numbers for which the sum of the squares of the digits is a square
172 - Investigating numbers with few repeated digits
173 - Using up to one million tiles how many different "hollow" square laminae can be formed?
174 - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements
175 - Fractions involving the number of different ways a number can be expressed as a sum of powers of 2
176 - Right-angled triangles that share a cathetus
177 - Integer angled Quadrilaterals
178 - Step Numbers
179 - Consecutive positive divisors
180 - Rational zeros of a function of three variables
181 - Investigating in how many ways objects of two different colours can be grouped
182 - RSA encryption
183 - Maximum product of parts
184 - Triangles containing the origin
185 - Number Mind
186 - Connectedness of a network
187 - Semiprimes
188 - The hyperexponentiation of a number
189 - Tri-colouring a triangular grid
190 - Maximising a weighted product
191 - Prize Strings
192 - Best Approximations
193 - Squarefree Numbers
194 - Coloured Configurations
195 - Inscribed circles of triangles with one angle of 60 degrees
196 - Prime triplets
197 - Investigating the behaviour of a recursively defined sequence
198 - Ambiguous Numbers
199 - Iterative Circle Packing
200 - Find the 200th prime-proof sqube containing the contiguous sub-string "200"
201 - Subsets with a unique sum
202 - Laserbeam
203 - Squarefree Binomial Coefficients
204 - Generalised Hamming Numbers
205 - Dice Game
206 - Concealed Square
207 - Integer partition equations
208 - Robot Walks
209 - Circular Logic
210 - Obtuse Angled Triangles
211 - Divisor Square Sum
212 - Combined Volume of Cuboids
213 - Flea Circus
214 - Totient Chains
215 - Crack-free Walls
216 - Investigating the primality of numbers of the form 2n2-1
217 - Balanced Numbers
218 - Perfect right-angled triangles
219 - Skew-cost coding
220 - Heighway Dragon
221 - Alexandrian Integers
222 - Sphere Packing
223 - Almost right-angled triangles I
224 - Almost right-angled triangles II
225 - Tribonacci non-divisors
226 - A Scoop of Blancmange
227 - The Chase
228 - Minkowski Sums
229 - Four Representations using Squares
230 - Fibonacci Words
231 - The prime factorisation of binomial coefficients
232 - The Race
233 - Lattice points on a circle
234 - Semidivisible numbers
235 - An Arithmetic Geometric sequence
236 - Luxury Hampers
237 - Tours on a 4 x n playing board
238 - Infinite string tour
239 - Twenty-two Foolish Primes
240 - Top Dice
241 - Perfection Quotients
242 - Odd Triplets
243 - Resilience
244 - Sliders
245 - Coresilience
246 - Tangents to an ellipse
247 - Squares under a hyperbola
248 - Numbers for which Euler’s totient function equals 13!
249 - Prime Subset Sums
250 - 250250
251 - Cardano Triplets
252 - Convex Holes
253 - Tidying up
254 - Sums of Digit Factorials
255 - Rounded Square Roots
256 - Tatami-Free Rooms
257 - Angular Bisectors
258 - A lagged Fibonacci sequence
259 - Reachable Numbers
260 - Stone Game
261 - Pivotal Square Sums
262 - Mountain Range
263 - An engineers' dream come true
264 - Triangle Centres
265 - Binary Circles
266 - Pseudo Square Root
267 - Billionaire
268 - Counting numbers with at least four distinct prime factors less than 100
269 - Polynomials with at least one integer root
270 - Cutting Squares
271 - Modular Cubes, part 1
272 - Modular Cubes, part 2
273 - Sum of Squares
274 - Divisibility Multipliers
275 - Balanced Sculptures
276 - Primitive Triangles
277 - A Modified Collatz sequence
278 - Linear Combinations of Semiprimes
279 - Triangles with integral sides and an integral angle
280 - Ant and seeds
281 - Pizza Toppings
282 - The Ackermann function
283 - Integer sided triangles for which the area/perimeter ratio is integral
284 - Steady Squares
285 - Pythagorean odds
286 - Scoring probabilities
287 - Quadtree encoding (a simple compression algorithm)
288 - An enormous factorial
289 - Eulerian Cycles
290 - Digital Signature
291 - Panaitopol Primes
292 - Pythagorean Polygons
293 - Pseudo-Fortunate Numbers
294 - Sum of digits - experience #23
295 - Lenticular holes
296 - Angular Bisector and Tangent
297 - Zeckendorf Representation
298 - Selective Amnesia
299 - Three similar triangles
300 - Protein folding
301 - Nim
302 - Strong Achilles Numbers
303 - Multiples with small digits
304 - Primonacci
305 - Reflexive Position
306 - Paper-strip Game
307 - Chip Defects
308 - An amazing Prime-generating Automaton
309 - Integer Ladders
310 - Nim Square
311 - Biclinic Integral Quadrilaterals
312 - Cyclic paths on Sierpiński graphs
313 - Sliding game
314 - The Mouse on the Moon
315 - Digital root clocks
316 - Numbers in decimal expansions
317 - Firecracker
318 - 2011 nines
319 - Bounded Sequences
320 - Factorials divisible by a huge integer
321 - Swapping Counters
322 - Binomial coefficients divisible by 10
323 - Bitwise-OR operations on random integers
324 - Building a tower
325 - Stone Game II
326 - Modulo Summations
327 - Rooms of Doom
328 - Lowest-cost Search
329 - Prime Frog
330 - Euler's Number
331 - Cross flips
332 - Spherical triangles
333 - Special partitions
334 - Spilling the beans
335 - Gathering the beans
336 - Maximix Arrangements
337 - Totient Stairstep Sequences
338 - Cutting Rectangular Grid Paper
339 - Peredur fab Efrawg
340 - Crazy Function
341 - Golomb's self-describing sequence
342 - The totient of a square is a cube
343 - Fractional Sequences
344 - Silver dollar game
345 - Matrix Sum
346 - Strong Repunits
347 - Largest integer divisible by two primes
348 - Sum of a square and a cube
349 - Langton's ant
350 - Constraining the least greatest and the greatest least
351 - Hexagonal orchards
352 - Blood tests
353 - Risky moon
354 - Distances in a bee's honeycomb
355 - Maximal coprime subset
356 - Largest roots of cubic polynomials
357 - Prime generating integers
358 - Cyclic numbers
359 - Hilbert's New Hotel
360 - Scary Sphere
361 - Subsequence of Thue-Morse sequence
362 - Squarefree factors
363 - Bézier Curves
364 - Comfortable distance
365 - A huge binomial coefficient
366 - Stone Game III
367 - Bozo sort
368 - A Kempner-like series
369 - Badugi
370 - Geometric triangles
371 - Licence plates
372 - Pencils of rays
373 - Circumscribed Circles
374 - Maximum Integer Partition Product
375 - Minimum of subsequences
376 - Nontransitive sets of dice
377 - Sum of digits, experience 13
378 - Triangle Triples
379 - Least common multiple count
380 - Amazing Mazes!
381 - (prime-k) factorial
382 - Generating polygons
383 - Divisibility comparison between factorials
384 - Rudin-Shapiro sequence
385 - Ellipses inside triangles
386 - Maximum length of an antichain
387 - Harshad Numbers
388 - Distinct Lines
389 - Platonic Dice
390 - Triangles with non rational sides and integral area
391 - Hopping Game
392 - Enmeshed unit circle
393 - Migrating ants
394 - Eating pie
395 - Pythagorean tree
396 - Weak Goodstein sequence
397 - Triangle on parabola
398 - Cutting rope
399 - Squarefree Fibonacci Numbers
400 - Fibonacci tree game
401 - Sum of squares of divisors
402 - Integer-valued polynomials
403 - Lattice points enclosed by parabola and line
404 - Crisscross Ellipses
405 - A rectangular tiling
406 - Guessing Game
407 - Idempotents
408 - Admissible paths through a grid
409 - Nim Extreme
410 - Circle and tangent line
411 - Uphill paths
412 - Gnomon numbering
413 - One-child Numbers
414 - Kaprekar constant
415 - Titanic sets
416 - A frog's trip
417 - Reciprocal cycles II
418 - Factorisation triples
419 - Look and say sequence
420 - 2x2 positive integer matrix
421 - Prime factors of n15+1
422 - Sequence of points on a hyperbola
423 - Consecutive die throws
424 - Kakuro
425 - Prime connection
426 - Box-ball system
427 - n-sequences
428 - Necklace of circles
429 - Sum of squares of unitary divisors
430 - Range flips
431 - Square Space Silo
432 - Totient sum
433 - Steps in Euclid's algorithm
434 - Rigid graphs
435 - Polynomials of Fibonacci numbers
436 - Unfair wager
437 - Fibonacci primitive roots
438 - Integer part of polynomial equation's solutions
439 - Sum of sum of divisors
440 - GCD and Tiling
441 - The inverse summation of coprime couples
442 - Eleven-free integers
443 - GCD sequence
444 - The Roundtable Lottery
445 - Retractions A
446 - Retractions B
447 - Retractions C
448 - Average least common multiple
449 - Chocolate covered candy
450 - Hypocycloid and Lattice points
451 - Modular inverses
452 - Long Products
453 - Lattice Quadrilaterals
454 - Diophantine reciprocals III
455 - Powers With Trailing Digits
456 - Triangles containing the origin II
457 - A polynomial modulo the square of a prime
458 - Permutations of Project
459 - Flipping game
460 - An ant on the move
461 - Almost Pi
462 - Permutation of 3-smooth numbers
463 - A weird recurrence relation
464 - Möbius function and intervals
465 - Polar polygons
466 - Distinct terms in a multiplication table
467 - Superinteger
468 - Smooth divisors of binomial coefficients
469 - Empty chairs
470 - Super Ramvok
471 - Triangle inscribed in ellipse
472 - Comfortable Distance II
473 - Phigital number base
474 - Last digits of divisors
475 - Music festival
476 - Circle Packing II
477 - Number Sequence Game
478 - Mixtures
479 - Roots on the Rise
480 - The Last Question
481 - Chef Showdown
482 - The incenter of a triangle
483 - Repeated permutation
484 - Arithmetic Derivative
485 - Maximum number of divisors
486 - Palindrome-containing strings
487 - Sums of power sums
488 - Unbalanced Nim
489 - Common factors between two sequences
490 - Jumping frog
491 - Double pandigital number divisible by 11
492 - Exploding sequence
493 - Under The Rainbow
494 - Collatz prefix families
495 - Writing n as the product of k distinct positive integers
496 - Incenter and circumcenter of triangle
497 - Drunken Tower of Hanoi
498 - Remainder of polynomial division
499 - St. Petersburg Lottery
500 - Problem 500!!!
501 - Eight Divisors
502 - Counting Castles
503 - Compromise or persist
504 - Square on the Inside
505 - Bidirectional Recurrence
506 - Clock sequence
507 - Shortest Lattice Vector
508 - Integers in base i-1
509 - Divisor Nim
510 - Tangent Circles
511 - Sequences with nice divisibility properties
512 - Sums of totients of powers
513 - Integral median
514 - Geoboard Shapes
515 - Dissonant Numbers
516 - 5-smooth totients
517 - A real recursion
518 - Prime triples and geometric sequences
519 - Tricoloured Coin Fountains
520 - Simbers
521 - Smallest prime factor
522 - Hilbert's Blackout
523 - First Sort I
524 - First Sort II
525 - Rolling Ellipse
526 - Largest prime factors of consecutive numbers
527 - Randomized Binary Search
528 - Constrained Sums
529 - 10-substrings
530 - GCD of Divisors
531 - Chinese leftovers
532 - Nanobots on Geodesics
533 - Minimum values of the Carmichael function
534 - Weak Queens
535 - Fractal Sequence
536 - Modulo power identity
537 - Counting tuples
538 - Maximum quadrilaterals
539 - Odd elimination
540 - Counting primitive Pythagorean triples
541 - Divisibility of Harmonic Number Denominators
542 - Geometric Progression with Maximum Sum
543 - Prime-Sum Numbers
544 - Chromatic Conundrum
545 - Faulhaber's Formulas
546 - The Floor's Revenge
547 - Distance of random points within hollow square laminae
548 - Gozinta Chains
549 - Divisibility of factorials
550 - Divisor game
551 - Sum of digits sequence
552 - Chinese leftovers II
553 - Power sets of power sets
554 - Centaurs on a chess board
555 - McCarthy 91 function
556 - Squarefree Gaussian Integers
557 - Cutting triangles
558 - Irrational base
559 - Permuted Matrices
560 - Coprime Nim
561 - Divisor Pairs
562 - Maximal perimeter
563 - Robot Welders
564 - Maximal polygons
565 - Divisibility of sum of divisors
566 - Cake Icing Puzzle
567 - Reciprocal games I
568 - Reciprocal games II
569 - Prime Mountain Range
570 - Snowflakes
571 - Super Pandigital Numbers
572 - Idempotent matrices
573 - Unfair race
574 - Verifying Primes
575 - Wandering Robots
576 - Irrational jumps
577 - Counting hexagons
578 - Integers with decreasing prime powers
579 - Lattice points in lattice cubes
580 - Squarefree Hilbert numbers
581 - 47-smooth triangular numbers
582 - Nearly isosceles 120 degree triangles
583 - Heron Envelopes
584 - Birthday Problem Revisited
585 - Nested square roots
586 - Binary Quadratic Form
587 - Concave triangle
588 - Quintinomial coefficients
589 - Poohsticks Marathon
590 - Sets with a given Least Common Multiple
591 - Best Approximations by Quadratic Integers
592 - Factorial trailing digits 2
593 - Fleeting Medians
594 - Rhombus Tilings
595 - Incremental Random Sort
596 - Number of lattice points in a hyperball
597 - Torpids
598 - Split Divisibilities
599 - Distinct Colourings of a Rubik's Cube
600 - Integer sided equiangular hexagons
601 - Divisibility streaks
602 - Product of Head Counts
603 - Substring sums of prime concatenations
604 - Convex path in square
605 - Pairwise Coin-Tossing Game
606 - Gozinta Chains II
607 - Marsh Crossing
608 - Divisor Sums
609 - π sequences
610 - Roman Numerals II
611 - Hallway of square steps
612 - Friend numbers
613 - Pythagorean Ant
614 - Special partitions 2
615 - The millionth number with at least one million prime factors
616 - Creative numbers
617 - Mirror Power Sequence
618 - Numbers with a given prime factor sum
619 - Square subsets
620 - Planetary Gears
621 - Expressing an integer as the sum of triangular numbers
622 - Riffle Shuffles
623 - Lambda Count
624 - Two heads are better than one
625 - Gcd sum
626 - Counting Binary Matrices
627 - Counting products
628 - Open chess positions
629 - Scatterstone Nim
630 - Crossed lines
631 - Constrained Permutations
632 - Square prime factors
633 - Square prime factors II
634 - Numbers of the form $a^2b^3$
635 - Subset sums
636 - Restricted Factorisations
637 - Flexible digit sum
638 - Weighted lattice paths
639 - Summing a multiplicative function
640 - Shut the Box
641 - A Long Row of Dice
642 - Sum of largest prime factors
643 - 2-Friendly
644 - Squares on the line
645 - Every Day is a Holiday
646 - Bounded Divisors
647 - Linear Transformations of Polygonal Numbers
648 - Skipping Squares
649 - Low-Prime Chessboard Nim
650 - Divisors of Binomial Product
651 - Patterned Cylinders
652 - Distinct values of a proto-logarithmic function
653 - Frictionless Tube
654 - Neighbourly Constraints
655 - Divisible Palindromes
656 - Palindromic sequences
657 - Incomplete words
658 - Incomplete words II
659 - Largest prime
660 - Pandigital Triangles
661 - A Long Chess Match
662 - Fibonacci paths
663 - Sums of subarrays
664 - An infinite game
665 - Proportionate Nim
666 - Polymorphic Bacteria
667 - Moving Pentagon
668 - Square root smooth Numbers
669 - The King's Banquet
670 - Colouring a Strip
671 - Colouring a Loop
672 - One more one
673 - Beds and Desks
674 - Solving $\mathcal{I}$-equations
675 - $2^{\omega(n)}$
676 - Matching Digit Sums
677 - Coloured Graphs
678 - Fermat-like Equations
679 - Freefarea
680 - Yarra Gnisrever
681 - Maximal Area
682 - 5-Smooth Pairs
683 - The Chase II
684 - Inverse Digit Sum
685 - Inverse Digit Sum II
686 - Powers of Two
687 - Shuffling Cards
688 - Piles of Plates
689 - Binary Series
690 - Tom and Jerry
691 - Long substring with many repetitions
692 - Siegbert and Jo
693 - Finite Sequence Generator
694 - Cube-full Divisors
← Problem 683
Inverse Digit Sum
Problem 685 →
Inverse Digit Sum
Problem 684
Define
to be the smallest number that has a digit sum of
. For example
.
Let
. You are given
.
Further let
be the Fibonacci sequence defined by
and
for all
.
Find
. Give your answer modulo
.