Lowest
common multiple and highest common factor
Key terms
The lowest common multiple (LCM) of two or more whole numbers is the lowest multiple shared by the numbers. For example, 4 and 5 have common multiples of 20, 40, 60, and so on. The lowest common multiple of 4 and 5 is 20.
The highest common factor (HCF) of two or more whole numbers is the highest factor shared by the numbers. For example, 16 and 24 have common factors 1, 2, 4 and 8. The highest common factor of 16 and 24 is 8.
A product is the result of multiplying two or more numbers. For example, the product of 2 and 3 is 6.
Worked
example 2
a) Find the lowest common multiple of 6 and 10.
b) Find the highest common factor of 18 and 27.
a) 6, 12, 18, 24, 30, 36 … 10, 20, 30, 40, 50 …
Lowest common multiple = 30
b) 18: 1, 2, 3, 6, 9, 18 27: 1, 3, 9, 27
Highest common factor = 9
Exercise 2
Write a list of multiples of 6, and a list of multiples of 10. Continue writing multiples until you find the smallest number that is in both lists. This is the lowest common multiple.
Write all the factors of 18 and all the factors of 27. The largest number that appears in both lists is the highest common factor.
Did you know?
A factor of a number is sometimes called a divisor. The highest common factor is sometimes called the greatest common divisor (GCD).
Think about
1 a) Which of these numbers are common multiples of 6 and 8?
b) Which of these numbers are common multiples of 10 and 15?
c) Which of these numbers are common multiples of 12 and 24?
How can you use a lowest common multiple to help you add the fractions 1 4 and 2 3 ?
Exercise 1
1 Which shape is not congruent to the others? Explain your answer.
2 In each of these sets of shapes, which one is not congruent to the others? Explain your answer.
Thinking and working mathematically activity
• Draw a rectangle ABCD and draw in the diagonal AC
Use a ruler and protractor to see if the two triangles ABC and CDA are congruent.
• Draw a parallelogram ABCD and draw in the diagonal BD.
Use a ruler and protractor to see if the two triangles BCD and ABD are congruent.
• Draw a parallelogram ABCD and draw in the diagonals AC and BD
Which triangles are congruent?
Explore whether congruent triangles are formed when a diagonal is drawn on other quadrilaterals.
Consolidation exercise
1 State whether or not the following pairs of triangles are congruent. Give reasons for your answers.
a)
2 A star is drawn inside a regular pentagon.
a) How many triangles are formed?
b) Name the triangles which are congruent to each other.
3 a) Draw a circle with radius 3.3 cm.
b) Draw a diameter AB.
c) Draw tangents to the circle at points A and B
d) What can you say about both tangents?
4 A pyramid has seven faces. How many vertices does it have?
5 What 3D shapes are being described below?
a) This shape has four faces and four vertices.
b) This shape has one face and no vertices.
c) This shape has seven faces, 15 edges and 10 vertices.
d) This shape has two faces that are octagons and eight faces that are rectangles.
6 The edges of a tetrahedron ABCD are all the same length.
a) Draw a sketch of the tetrahedron.
b) Name the congruent triangles found on the tetrahedron.
7 Six of the faces of a 3D shape are congruent to each other. The other two faces of this shape are also congruent to each other. Write down the name of this shape.
Exercise 2 1–8
1 The temperature changes by −2 °C every hour. What is the total change in temperature after 7 hours?
2 Find:
a) –4 × 4
b) 5 × (−7)
e) –3 × 6 f) 7 × 5
3 Find:
a) −4 ÷ 4
e) 32 ÷ 8
b) 20 ÷ (−2)
f) 45 ÷ (−9)
c) –5 × 9
d) –10 × 2
g) 8 × (−4) h) 5 × (−11)
c) −25 ÷ 5
d) −42 ÷ 7
g) 8 ÷ (−4) h) −28 ÷ 4
4 Write true or false for each calculation. If it is false, correct it.
a) −49 ÷ 7 = 7 b) –7 × 4 = −21
c) 80 ÷ (−10) = 8 d) 3 × (−2) = −6
5 Write two multiplication calculations and two division calculations that give each answer.
a) −10 b) 32
6 Find the missing numbers.
a) ____ × (−4) = −28 b) 6 × ____ = 30
c) −12
c) 22 ÷ ____ = −11
d) ____ ÷ (−5) = −6 e) 9 × ____ = −63 f) ____ ÷ 4 = −6
7 You are given the result −288 ÷ 24 = −12. Without doing any working, write at least six other calculations that must be true.
8 Below are two puzzles. Copy each puzzle and write numbers in the four boxes to make all of the calculations correct.
a) ×= − 8 ×÷ ×= 6 = 6= −2 b) ÷= 6 ÷× ÷= −2 = −3= 25
9 Use a calculator to find:
a) 38 × (−69)
d) 9126 ÷ (−3)
b) –87 × 6
e) –189 ÷ 3
Thinking and working mathematically activity
Tip
In question 7, can you write another division and a multiplication? If you change the signs on numbers, can you write more multiplications and divisions?
c) 4050 ÷ (−50)
f) –341 × 214
x and y are integers. Look for values of x and y that give each of the types of result below.
• x ÷ y is less than both x and y
• x ÷ y is greater than y.
• x ÷ y is greater than x.
• x ÷ y is greater than both x and y Write rules to summarise your findings. Try to explain why each rule works.