Number Toolkit

Integers and natural numbers

• The integers run in both directions from zero without fractional parts: … −3, −2, −1, 0, 1, 2, 3 …
• Counting from 1 upwards gives the natural numbers 1, 2, 3, 4, …, a set that leaves zero out
• A number written as a decimal or a fraction is only an integer if it simplifies to a whole number
  • 18/3 is an integer because it equals 6

Multiples

• A multiple of an integer is the result of multiplying it by a positive integer
• The multiples of 7 are 7, 14, 21, 28, 35, …
• A common multiple of two integers is a multiple of both
  • 24 is a common multiple of 6 and 8
• Multiples can be written algebraically: the multiples of n are n, 2n, 3n, 4n, …

Factors

• A factor of an integer divides into it exactly, leaving no remainder
• Beyond 1, no integer has fewer than two factors, since 1 and the number itself always divide it
• Factors are found in pairs, which is why testing only up to the square root is enough
  • For 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6, giving factors 1, 2, 3, 4, 6, 8, 12, 24
• A common factor of two integers is a factor of both
  • 5 is a common factor of 20 and 35

Divisibility tests

• Divisibility tests decide whether a number is a factor without dividing, which matters on the non-calculator papers

Primes, squares, cubes and roots

• A prime number has exactly two factors, 1 and itself
  • Among the primes only 2 is even, since every other even number has 2 as a third factor
  • 1 is not prime, because it has only one factor
• A square number is an integer multiplied by itself: 1, 4, 9, 16, 25, 36, …
• A cube number is an integer multiplied by itself twice: 1, 8, 27, 64, 125, …
• The square root of a number is the value that squares to give it, written √
  • √49 = 7, because 7² = 49
• The cube root reverses cubing, written ∛
  • ∛125 = 5, because 5³ = 125