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Prime Factors, HCF & LCM (Edexcel IGCSE Maths A)

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  • Define the term integer.

    Integers are whole numbers. They can be positive, negative or zero.

  • What is a factor?

    A factor of a number is a positive integer that divides exactly into the number.

  • What is a multiple?

    A multiple of a number is a positive integer that can be made by multiplying the number by another integer.

  • Define a prime number.

    A prime number is a positive integer which has exactly two factors, itself and 1.

  • True or False?

    1 is a prime number.

    False.

    1 is not a prime number, as it only has one factor.

  • What is the only even prime number?

    2 is the only even prime number.

  • What are the first ten prime numbers?

    The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

  • What are the first twelve square numbers?

    The first twelve square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

  • What are the first five cube numbers?

    The first five cube numbers are: 1, 8, 27, 64, 125.

  • True or False?

    Any positive integer has an infinite number of multiples.

    True.

    Any positive integer has an infinite number of multiples.

  • What is a natural number?

    A natural number is a positive integer.

    They are often referred to as counting numbers.
    Zero is not a natural number.

  • True or False?

    The result of any number multiplied by its reciprocal will be equal to 1.

    True.

    The result of any number multiplied by its reciprocal will be equal to 1.

    E.g. 3 cross times 1 third equals 1

  • What are prime factors?

    Prime factors of a number are the prime numbers which are factors of the number.

    E.g. 24 has the prime factors 2 and 3, (24 = 23 x 3)

  • What is prime factor decomposition?

    Prime factor decomposition (or Prime Factorisation) is the process of breaking a number up into its prime factors.

    Break a number into a pair of factors, then break those factors down in pairs in the same way until you are left with only prime factors.

    table row 72 equals cell 8 cross times 9 end cell row blank equals cell open parentheses 2 cross times 4 close parentheses cross times open parentheses 3 cross times 3 close parentheses end cell row blank equals cell 2 cross times open parentheses 2 cross times 2 close parentheses cross times 3 cross times 3 end cell row blank bold equals cell bold 2 bold cross times bold 2 bold cross times bold 2 bold cross times bold 3 bold cross times bold 3 end cell end table

  • How do you write a number as a product of prime factors?

    A product of prime factors can be found using prime factor decomposition then writing the prime factors of a number multiplied together.

    E.g. 36 equals 2 cross times 2 cross times 3 cross times 3 or 36 equals 2 squared cross times 3 squared.

  • True or False?

    500 can be written as a product of prime factors in the form 2 cross times 2 cross times 5 cross times 5 cross times 5.

    True.

    The prime factorisation of 500 equals 2 cross times 2 cross times 5 cross times 5 cross times 5.

    You may be asked to give your answer in the form 500 equals 2 squared cross times 5 cubed.

  • How can prime factor decomposition be used to identify if a number is a square number?

    To identify if a number is a square number, use prime factor decomposition.
    If the indices of all of its prime factors are even, the number is square.

    E.g. 36 is a square number: 36 equals 2 squared cross times 3 squared

    24 is not a square number: 24 equals 2 cubed cross times 3

  • True or False?

    You can identify a cube number by the fact that the index of each its prime factors is a cube.

    False.

    You can identify a cube number by the fact that the index of each of its prime factors is a multiple of 3.

    E.g. 216 is a cube number: 216 equals 2 cubed cross times 3 cubed

  • True or False?

    It can be shown, using its prime factors, that 10 square root of 15 is the exact square root of 1500.

    True.

    First write 1500 as a product of its prime factors, e.g. 1500 equals 2 squared cross times 3 cross times 5 cubed

    Separate prime factors with an even index and those with an odd index, e.g. 1500 equals open parentheses 2 squared cross times 5 squared close parentheses cross times open parentheses 3 cross times 5 close parentheses

    Take the square root of both groups,
    e.g. table row cell square root of 1500 end cell equals cell square root of open parentheses 2 squared cross times 5 squared close parentheses end root cross times square root of open parentheses 3 cross times 15 close parentheses end root end cell row blank equals cell 2 cross times 5 cross times square root of 15 end cell row blank equals cell 10 square root of 15 end cell end table

  • What is a common factor?

    A common factor of two numbers is a positive integer that both numbers can be divided by exactly.

    E.g. 3 is a common factor of the numbers 6 and 21.

  • What does the term highest common factor (HCF) of two numbers mean?

    The highest common factor (HCF) is the largest positive integer that divides into both numbers exactly.

    E.g. 5 is the highest common factor of the numbers 10 and 15.

  • What number is always a common factor of any two positive integers?

    1 is always a common factor of any two numbers.

  • How can you find the HCF of two numbers?

    To find the HCF, write each number as a product of its prime factors, find the common prime factors, and multiply them together.

    E.g. To find the HCF of the numbers 45 and 60

    Each number as a product of its prime factors is: 45 equals 3 squared cross times 5, 60 equals 2 squared cross times 3 cross times 5

    The common prime factors are: 3 and 5

    The HCF is therefore: 3 cross times 5 equals 15

  • What does the term lowest common multiple (LCM) of two numbers mean?

    The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

    E.g. The LCM of 4 and 6 is 12.

  • What is a common multiple?

    A common multiple of two numbers is a number that appears in both of their times tables.

    E.g. Common multiples of the numbers 3 and 5 include 15, 30, 15, ...etc.

  • True or False?

    The product of two numbers is always the lowest common multiple.

    False.

    The product of two numbers is always a common multiple, however it is not always the smallest.

  • How can you find the LCM of two numbers?

    To find the LCM, write each number as a product of the powers of its prime factors. Then find the highest power of every prime that appears in either number.
    The lowest common multiple is the product of these highest powers.

    E.g. To find the LCM of 108 and 126
    108 equals 2 squared cross times 3 cubed and 126 equals 2 cross times 3 squared cross times 7
    Multiply together the highest powers of all the primes: 2 squared cross times 3 cubed cross times 7 to the power of 1 equals bold 756

  • True or False?

    Any pair of numbers has a finite number of common factors.

    True.

    Any pair of numbers will always have a finite number of common factors.

  • True or False?

    Any pair of numbers has a finite number of common multiples.

    False.

    Any pair of numbers will always have an infinite number of common multiples.

    Once you have found a common multiple, you can multiply that by any positive integer to find another common multiple.

Previous:Set Notation & Venn DiagramsNext:Powers, Roots & Standard Form

    1. Topic 1

      Number Toolkit

    2. Topic 2

      Set Notation & Venn Diagrams

    3. Topic 3

      Prime Factors, HCF & LCM

    4. Topic 4

      Powers, Roots & Standard Form

    5. Topic 5

      Fractions

    6. Topic 6

      Percentages

    7. Topic 7

      Compound Interest & Depreciation

    8. Topic 8

      Fractions, Decimals & Percentages

    9. Topic 9

      Rounding, Estimation & Bounds

    10. Topic 10

      Surds

    11. Topic 11

      Using a Calculator

    12. Topic 12

      Ratio Toolkit

    13. Topic 13

      Ratio Problem Solving

    14. Topic 14

      Exchange Rates & Best Buys

    15. Topic 15

      Direct & Inverse Proportion

    1. Topic 1

      Algebra Toolkit

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Solving Linear Equations

    10. Topic 10

      Solving Quadratic Equations

    11. Topic 11

      Solving Inequalities

    12. Topic 12

      Simultaneous Equations

    13. Topic 13

      Forming & Solving Equations

    1. Topic 1

      Sequences

    2. Topic 2

      Functions

    3. Topic 3

      Coordinate Geometry

    4. Topic 4

      Linear Graphs y = mx + c

    5. Topic 5

      Graphs of Functions

    6. Topic 6

      Estimating Gradients

    7. Topic 7

      Real-Life Graphs

    8. Topic 8

      Graphing Inequalities

    9. Topic 9

      Transformations of Graphs

    10. Topic 10

      Differentiation

    1. Topic 1

      Standard & Compound Units

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing & Constructions

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Right-Angled Triangles - Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    1. Topic 1

      Vectors

    2. Topic 2

      Transformations

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Histograms

    3. Topic 3

      Cumulative Frequency Diagrams

    4. Topic 4

      Probability Toolkit

    5. Topic 5

      Probability Diagrams - Venn & Tree Diagrams

    6. Topic 6

      Combined & Conditional Probability