GCSE Maths

Numbers
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Integers
Prime Numbers and Composite Numbers
Negative Numbers
Comparing Numbers Using Symbols
The Square Numbers
The Laws of Indices
Negative Indices
The Cube Numbers
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
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Vectors
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Vectors
Probability
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Introduction to probability
Finding the sum of probabilities
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
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Population and samples
Representing data
Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
Quadratic Equations
Mathematical Formulae
Algebraic Equivalence and Proof
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Generating sequences
Nth Term Sequences
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Inverse Proportion
Conversion
Scale Factors
Percentages
Direct Proportion
Rate, Ratio and Proportion Examples
Ratios
Ratios
Ratio, Proportion and Rates of Change
Geometry and Measures
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Properties of angles
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Transformations
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Plane figure properties
Plans and elevations
Areas and Volumes in Map Scale Problems
Mensuration and Calculation
10 Topics
Map scales
Bearings
Area
Volume
Circles: arcs and sectors
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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The Highest Common Factor

GCSE Maths Numbers The Highest Common Factor

The highest common factor (HCF) of two or more numbers is the largest number that is a commo factor of the numbers.
To find it, write down the numbers as their prime factors. Then choose those factors that appear in all of the numbers.

Let’s look at some examples.

Example

Find the HCF of the numbers 15, 24 and 30.

15=3\times 5
24=2\times 2\times 2\times 3
30=2\times 2\times 5
We see that 3 is common in them all. So the HCF is 3.

Example

Find the HCF pf the numbers 16, 24, 72 and 80.

Here 16=2\times 2\times 2\times 2
124=2\times 2\times 2\times 2\times 3
72=2\times 2\times 2\times 3\times 3
80=2\times 2\times 2\times 2\times 5
We see that 2\times 2\times 2 appears in them all, hence the HCF is 8.
Even with bigger numbers we proceed in the same manner.

Example

Find the HCF of 1245 and 3000.

To find the prime factors, we use the method of long division.
3|1245
5|415
83|83
1
And
2|3000
2|2500
2|750
3|375
5|125
5|25
5|5
1
So, we have 1245=3\times 5\times 83 and
3000=2\times 2\times 2\times 3\times 5\times 5\times 5
This means 3\times 5 is common in both, therefore HCF is 3\times 5=15

Example

The next example is taken from a Cambridge past examination paper.

The numbers 168 and 324, written as the products of their prime factors are

168={ 2 }^{ 2 }\times 3\times 7

324={ 2 }^{ 2 }\times { 3 }^{ 4 }

Find:

1. \sqrt { 324 }

2. The largest integer which is a factor of both 168 and 324.

3. The smallest positive integer value of n for which 168n is a multiple of 324.

1. \sqrt { 324 } =\sqrt { { 2 }^{ 2 }\times { 3 }^{ 4 } }

{ 2 }^{ 1 }\times { 3 }^{ 2 }

=2\times 9

=18

2. We see { 2 }^{ 2 }\times 3=12

3. Compare 168={ 2 }^{ 3 }\times 3\times 7 and 324={ 2 }^{ 2 }\times { 3 }^{ 4 }

To find the smallest positive integer value of n for which 168n is a multiple of 324.

We write 168n={ 2 }^{ 3 }\times 3\times 7\times n

To be a multiple of 324={ 2 }^{ 2 }\times { 3 }^{ 4 } should be a factor of { 2 }^{ 3 }\times 3\times 7\times n and

\frac { 168n }{ 324 } \quad = a whole number,

\frac { { 2 }^{ 3 }\times 3\times 7\times n }{ { 2 }^{ 2 }\times { 3 }^{ 4 } } \quad = a whole number

\frac { 2\times 7\times n }{ { 3 }^{ 3 } } \quad = a whole number

So that, we should choose n={ 3 }^{ 2 }=27 to make the division a whole number answer.

Example

Find the H.C.F of 210,315 and 650.

We first find the prime factors of 210:

2|210

3|105

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5|65
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HCF of 210, 315 and 650 = 5$

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