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Number Theory

Map

Arrows point from what you learn first to what it opens. Right-click a concept (or press and hold it) for its lessons and stories.

  • Proven to depth123
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Clock arithmeticDivisibilityDivisibility testsFactorsGreatest common factorLeast common multipleMultiplesPrime factorisationPrime numbersSquare numbersTriangular numbersMultiplicationIn ArithmeticPowersIn ArithmeticSquare rootsIn ArithmeticAdding fractionsIn FractionsDivisionIn ArithmeticPlace valueIn Number SystemsAdditionIn ArithmeticConjecturesIn Sets and Proof

Learning path

In the order the concepts build on each other.

  1. Clock arithmeticLockedCounting that goes round in a circle, like the hours on a clock or the days of a week: on a clock 9 + 5 = 2. Full turns bring you back where you started, so only the remainder after dividing by the size of the circle matters. Mathematicians call it modular arithmetic.Opens after: Division
  2. MultiplesLockedThe multiples of a number are where you land counting in jumps of it: the multiples of 6 are 0, 6, 12, 18, 24, ... They never end, 0 is one of them, and a number is a multiple when dividing it leaves nothing over.Opens after: Multiplication
  3. Square numbersLockedA square number is a number of dots that can be laid out as a perfect square: 1, 4, 9, 16, 25 ... Each is a whole number times itself, like 16 = 4 × 4, and each is the one before plus the next odd number.Opens after: Multiplication
  4. FactorsLockedThe factors of a number are the row lengths that make a rectangle of it: 12 chairs make rows of 1, 2, 3, 4, 6 or 12, so those are the factors of 12. They come in pairs (3 × 4 = 12), and the pairs meet in the middle.Opens after: Multiples
  5. Least common multipleLockedThe smallest number that is a multiple of two numbers. A bus every 4 minutes and a bus every 6 minutes leave together again after 12 minutes, the first number in both lists of multiples. It is the product of the two numbers only when they share no factor but 1. Also called the lowest common multiple.Opens after: Multiples
  6. Triangular numbersLockedA triangular number is a number of dots that can be laid out as a triangle with rows of 1, 2, 3 and so on: 1, 3, 6, 10, 15 ... Two copies of a triangle make a rectangle, so 1 + 2 + ... + 100 is 100 × 101 ÷ 2 = 5,050, and two neighbouring triangular numbers make a square.Opens after: Addition, Square numbers
  7. DivisibilityLockedOne number divides another when sharing leaves nothing over: 4 divides 24, so 24 is divisible by 4. "3 is a factor of 24", "24 is a multiple of 3" and "3 divides 24" are the same fact said three ways.Opens after: Factors, Division
  8. Greatest common factorLockedThe biggest number that divides two numbers exactly. 12 biscuits and 18 sweets make at most 6 equal party bags, because 6 is the biggest number in both lists of factors (12 = 2 × 2 × 3 and 18 = 2 × 3 × 3 share 2 × 3). Also called the highest common factor.Opens after: Factors
  9. Prime numbersLockedNumbers bigger than 1 whose only factors are 1 and themselves - the numbers whose dots make only a single line, never a rectangle.Opens after: Factors
  10. Divisibility testsLockedQuick ways to tell whether a number divides by another without dividing. For 2, 5 and 10 the last digit decides, because every ten already splits into twos, fives and tens. For 3 and 9 the digit sum decides, because every ten is a nine plus one.Opens after: Divisibility, Place value
  11. Prime factorisationLockedWriting a number as primes multiplied together, like 60 = 2 × 2 × 3 × 5 - the one recipe every whole number bigger than 1 has.Opens after: Prime numbers

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