Prime factors, HCF & LCM

• What vocabulary related to types of numbers will you encounter at IGCSE level?

  Real numbers, integers, natural numbers, indices, factors, multiples, prime, square, and cube numbers
• What are real numbers?

  Real numbers are the set of all numbers, including integers, fractions, rational and irrational numbers
• What symbol is used to denote real numbers?


  The symbol ℝ
• What are integers?

  Integers are all whole numbers, which can be positive, negative, or zero
• What symbol is used to denote integers?


  The symbol ℤ
• What are natural numbers?

  Natural numbers are the set of all positive integers, often thought of as counting numbers
• What symbol is used to denote natural numbers?


  The symbol ℕ
• What is a factor?

  A factor is a number that divides into another number exactly
• What are the factors of 18?

  The factors of 18 are 1, 2, 3, 6, 9, and 18
• What is a multiple?

  A multiple is a number that is in the times table of another number
• What are the multiples of 3?

  The multiples of 3 are 3, 6, 9, 12, 15, 18, and so on
• What is a prime number?

  A prime number is a number which has exactly two factors, itself and 1
• Why is 1 not considered a prime number?

  1 is not a prime number because it only has one factor
• What is the only even prime number?

  2
• What are the first ten prime numbers?

  2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• What is a square number?


  A square number is the number derived from multiplying a number by itself
• What is an example of a square number?

  9, because 3 × 3 = 9
• What are the first twelve square numbers?

  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
• What is a cube number?

  A cube number is the number derived from multiplying a number by itself twice
• What is an example of a cube number?

  27, because 3 × 3 × 3 = 27
• What are the first five cube numbers?

  1, 8, 27, 64, 125
• What is an index in mathematics?

  An index is a way of writing a string of multiplications of the same number neatly
• What is the plural of index?

  Indices
• How can the expression \(3 \times 3 \times 3 \times 3\) be written in index form?


  It can be written as \(3^4\)
• How can the expression \(a \times a \times a \times a \times b \times b \times b \times b \times b\) be written in index form?


  It can be written as \(a^4 \times b^5\)
• What are prime factors?

  Prime factors are all the prime numbers which multiply to give a number
• What are the first few prime numbers?

  2, 3, 5, 7, 11, 13, 17, 19, …
• How do you find the prime factors of a number?

  Use a factor tree to split a number into a pair of factors until you reach a prime number
• What is the unique representation of a number in terms of prime factors?

  A number can be uniquely written as a product of prime factors
• How should prime factors be written?

  Prime factors should be written in ascending order with \( \times \) between them
• How can you express a number as the product of prime factors?

  By using phrases like "Express ... as the product of prime factors" or "Find the prime factor decomposition of ..."
• What is the prime factor decomposition of 360?

  The prime factor decomposition of 360 is \(2^3 \times 3^2 \times 5^1\)
• How can prime factor decomposition (PFD) be used to determine if a number is a square number?

  If all the indices in the prime factor decomposition are even, then that number is a square number
• How can PFD be used to determine if a number is a cube number?

  If all the indices in the prime factor decomposition are multiples of 3, then that number is a cube number
• How can you find the square root of a square number using PFD?

  Write the number in its prime factor decomposition, halve all the indices, and multiply the prime factors together
• What are the steps to find the exact square root of a number using its prime factors?


  1. Write the prime factors out as individual factors
  2. Pair the factors together
  3. Find the product of each paired prime factor
  4. Multiply the remaining factors together
  5. Write the answer as a product of the integer and the square root of the remaining factors

  
• What is the highest common factor (HCF) of two numbers?

  The highest common factor is the largest common factor between two numbers
• How do you find the HCF of two numbers?

  Write each number as a product of its prime factors and multiply the common prime factors together
• What is the lowest common multiple (LCM) of two numbers?

  The lowest common multiple is the smallest common multiple between two numbers
• How do you find the LCM of two numbers?

  Write each number as a product of its prime factors and multiply the first number by the extra prime factors