One-page revision sheet

• Integers are the positive and negative whole numbers together with zero: … −3, −2, −1, 0, 1, 2, 3 …
• Adding a positive number moves you right on the number line; subtracting a positive number moves you left.
• Adding a negative is the same as subtracting: a + (−b) = a − b.
• Subtracting a negative is the same as adding: a − (−b) = a + b.
• The further left a number is on the number line, the smaller it is, so −7 < −3 < 0 < 2.

Key words: integer  ·  negative number  ·  inverse  ·  inverse operation  ·  sum  ·  difference

• positive × positive = positive   e.g. 6 × 7 = 42
• positive × negative = negative   e.g. 9 × (−4) = −36
• negative × negative = positive   e.g. (−8) × (−5) = 40
• The same three rules apply to division.
• In a chain of multiplications, count the negative factors: an even number of negatives gives a positive answer, an odd number gives a negative answer.
• Watch the brackets: (−4)² = 16 but −4² = −16.

Key words: product  ·  factor  ·  quotient  ·  index  ·  inverse operation

• The multiples of a number are its times table: multiples of 4 are 4, 8, 12, 16, 20, …
• A common multiple of two numbers appears in both times tables.
• The lowest common multiple (LCM) is the smallest of those common multiples.
• Method: list the multiples of the larger number and stop at the first one that the smaller number divides into.
• If two numbers share no factors, LCM = a × b, e.g. LCM(5, 8) = 40.
• If one number is a multiple of the other, the LCM is the larger number, e.g. LCM(4, 12) = 12.

Key words: multiple  ·  common multiple  ·  lowest common multiple (LCM)  ·  digit

• A factor divides exactly into a number: the factors of 12 are 1, 2, 3, 4, 6, 12.
• A common factor divides into both numbers.
• The highest common factor (HCF) is the biggest of them.
• Method: list the factors of the smaller number, then work downwards and stop at the first one that also divides the larger number.
• If one number is a factor of the other, the HCF is the smaller number, e.g. HCF(6, 18) = 6.
• Useful fact: HCF × LCM = a × b, e.g. for 8 and 12: 4 × 24 = 96 = 8 × 12.
• Dividing a fraction's numerator and denominator by their HCF simplifies it in one step.

Key words: factor  ·  common factor  ·  highest common factor (HCF)  ·  co-prime  ·  simplify

• ÷ 2 — the last digit is even
• ÷ 3 — the sum of the digits is divisible by 3
• ÷ 4 — the number formed by the last two digits is divisible by 4
• ÷ 5 — the last digit is 0 or 5
• ÷ 6 — it passes the tests for both 2 and 3
• ÷ 8 — the number formed by the last three digits is divisible by 8
• ÷ 9 — the sum of the digits is divisible by 9
• ÷ 10 — the last digit is 0
• ÷ 11 — the difference between the sum of the odd-position digits and the sum of the even-position digits is 0 or a multiple of 11
• You can combine tests when the two divisors share no factors: a number divisible by 3 and by 4 is divisible by 12. But 2 and 4 do share a factor, so passing both does not guarantee divisibility by 8.

Key words: divisible  ·  tests of divisibility  ·  remainder  ·  digit sum  ·  consecutive

• Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, …
• Cube numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
• √ undoes squaring, and ∛ undoes cubing — they are inverse operations.
• √144 = 12 because 12² = 144; ∛125 = 5 because 5³ = 125.
• To estimate √n when n is not a square number, find the square numbers either side of it: 49 < 50 < 64, so 7 < √50 < 8.
• 64 is special — it is both a square number (8²) and a cube number (4³).

Key words: square number  ·  square root  ·  cube number  ·  cube root  ·  index  ·  estimate