Arithmetic

Subcategories

• Clock10 +4
• Digits73 +40
• Tests for divisibility5 +2

Puzzles

• 7-11A customer at a 7-11 store selected four items to buy, and was told that the cost was $7.11. Added material
• aliquot.sequencesIn an aliquot chain one starts with a number, calculates the sum of its proper divisors, and repeats the process on the result. New
• arithmetic.progressionIs there an arithmetic progression of 20 or more primes?
• birds.in.treesThere is a field. New, unsolved
• cattle.of.the.sunArchimedes' cattle problem has two versions, based on alternate readings of its second paragraph; what are... New, unsolved
• charred.willAfter a fire destroyed the home, the life and the will of a rich aristocrat, a detective found among the... New, unsolved
• coconutsFive sailors collect a pile of coconuts; in the night each in turn divides the pile into five, gives the... New
• collatzStart with any positive integer x; if it is even, replace it by x/2, if odd by 3x+1, and repeat. New
• consecutive.compositesAre there 10,000 consecutive non-prime numbers?
• consecutive.p2qNumbers of the form (p^2)q, p and q different primes, are "special". New
• consecutive.productProve that the product of three or more consecutive positive integers cannot be a perfect square.
• consecutive.sumsFind all series of consecutive positive integers whose sum is exactly 10,000.
• conwayDescribe the sequence a(1)=a(2)=1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. Added material
• count.one.bitsCount the number of ones in a binary number using as few instructions as possible. New
• dogs.meadSolve the cross-number puzzle "Dog's Mead" (also known as "Little Pigley Farm"), in which every entry is a... New, unsolved
• dogs.mice.catsA dog costs $15, four mice cost $1 and a cat costs $1. New
• drawing.pixelsA computer draws a horizontal line 128 pixels long in n seconds with the command "Plot 0,0 128,0 n". New
• egyptian.fractionCan you represent 36/457 as a sum of different fractions with numerator 1? New
• egyptian.fractionsExpress 36/457 as a sum of distinct unit fractions (fractions with numerator 1). New
• element.alphameticEach of the 26 letters has a secret numerical value, and the value of a word is the sum of the values of its letters. New
• factoring.recordWhat is the largest number factored by a general purpose algorithm, and how was it done? New
• fermat.numbersWhat factors are known of the Fermat numbers F_n = 2^(2^n) + 1? New
• feynman.fermatIs there a probabilistic argument that Fermat's Last Theorem is true? New
• five.five.fiveMake the equation 5+5+5=550 true by adding just a single straight line. New
• four.squaresFind integers a, b such that a+45b, a+5b, a+15b and a-3b are all squares. New
• fourth.powersEuler conjectured in 1769 that a^4 + b^4 + c^4 = d^4 has no solutions in positive integers. New
• goldbachWhat is known about Goldbach's conjecture, that every even integer greater than 4 is the sum of two odd... New
• grandpa.sonsWhen Grandpa was over fifty but less than eighty, he told to a friend: "each of my sons has as many sons as... New
• hamming.numbersHamming numbers are defined as numbers whose only prime factors are 2, 3, and 5. New
• highly.compositeWhich numbers n have more divisors than any smaller number, and how large is the number of divisors d(n)... New
• house.numbersA street has between 50 and 500 houses numbered 1, 2, 3, ..., N, and there is a house X such that the sum... New
• integral.rootsWhat is the second smallest integer with integral third, fourth and fifth roots? New
• irrational.powerCan you prove that there is at least one instance where an irrational number raised to an irrational power... New
• knapsacksOn the last day of his vacation, a man goes into a gift shop to buy presents for his grandchildren. New
• largest.non.mersenne.primeWhat is the largest prime integer known to man that is NOT one less than a power of two? New
• lockersThere are 100 closed lockers in a row, numbered 1 to 100, and 100 students. New
• mad.veterinarianA mad veterinarian has created three animal transmogrifying machines. New
• magic.hexagramThe Magic Hexagon New
• magic.squaresAre there large squares, containing only consecutive integers, all of whose rows, columns and diagonals have the same sum? Added material
• max.ageMax discovered that if he subtracted 7 from his age and multiplied it by 7, he got the same answer as... New
• mersenne.primesWhat Mersenne primes 2^p - 1 are known, and when, how and by whom was each one found? New
• negative.baseHow much is 7 in base (-1)? New
• nth.primeThe 1st prime is 2, the 10th prime is 29, the 100th prime is 541, the 1000th prime is 7919, the 10000th... New
• pellFind integer solutions to x^2 - 92y^2 = 1.
• perfect.numbersWhat are all [ Editor: "Can you determine all" in the original ] the "perfect" numbers between 1 and 1,000,000? New
• periodic.recurrenceCan a non-trivial integer sequence in which each element is the sum of the k previous elements be periodic? New
• prime.magic.squareIs there a 3x3 magic square made up of nine different prime numbers? New
• prime.products.setShow that there exists a set A of positive integers with the following property: for every infinite set S... New
• prime.seriesHere is a number series: 41, 43, 47, 53, ... New
• prime.square.sumFind all solutions in integers to a^2 + b^2 + c^2 = d^2, where a, b and d are prime. New
• product.of.powersFind rational numbers a,b,c,d,e,f,g,h,i,j such that: New
• pythagorean.odd.legsIs it possible to construct a right triangle with integer sides having both of the legs odd integers? New
• ratio.64.7The product of some of the numbers 1 to 10 was divided by the product of the remaining numbers. New
• rational.rootsShow that an n-th root of an integer is either an integer or irrational. New
• remainder.one.lessWhat number, if divided by 10, leaves a remainder of 9; then divided by 9 leaves a remainder of 8; divided... New
• repeated.integersCome up with the general formula for the n-th term of the following sequence: New
• seven.unit.fractionsFind seven different unit fractions whose sum is 1. New
• snails.crossingA family of snails takes one hour to cross a lawn, from the time that the first snail puts its 'nose' onto... New
• square.productsProve that for any three positive integers a, b, c such that the product of any two is one less than a square, New
• square.progression.chainIt is not uncommon to find a triple of integers whose squares are in arithmetic progression, e.g. New
• square.root.partitionSplit the square roots of 1, 2, 3, ..., 49, 50 into two groups so that the difference between the sums of... New
• subsetProve that all sets of n integers contain a subset whose sum is divisible by n.
• sum.of.cubesFind two fractions whose cubes total 6.
• sum.of.two.primesIs every even number greater than 2 the sum of two primes? New
• sums.of.powersPartition 1,2,3,...,16 into two equal sets, such that the sums of the numbers in each set are equal, as are...
• taxicabIs 1729 the only number that is the sum of two cubes in two different ways? New
• tenth.root.of.tenWhich is larger, the tenth root of ten or the cube root of two? New
• three.n.plus.oneWhat is a good reference on the 3N+1 problem? New
• titanic.primesWhat are the known titanic primes (primes with at least 1000 decimal digits) that are not Mersenne primes? New
• trampsFour tramps divide 200 hours of work so that each works a different number of hours a day, for as many days as he works hours a day. New
• truncated.multiplicationGiven a function h on {0,1,...,m}, is it a composition of "truncated multiplications" f(x) = floor(x*n)... New
• two.squares.odd.powerHow many solutions are there to a^2 + b^2 = c^5 for integers a, b, c? New
• upside.down.magic.squareUsing the sixteen two-digit numbers made from the digits 1, 6, 8 and 9 (11, 16, 18, 19, 61, 66, 68, 69, 81,... New