IGCSE Mathematics (0580)
Chapter 1: Numbers
Positive and negative integers, natural numbers, rational numbers, real numbers and
their four operations
The set of natural numbers, = {1, 2, 3, ….}
The set of integers, = {…, -2, -1, 0, 1, 2, ….}
The set of positive integers, + = {1, 2, 3, ….}
The set of negative integers, - = {-1, -2, -3…}
The set of rational numbers are numbers which can be expressed in the form
b
,
a
where a and b are integers, and b is not equal to 0.
An irrational number is a number which cannot be expressed in the form
Irrational numbers include √2, π, and e.
The set of real numbers is the set of rational and irrational numbers.
Prime numbers are numbers that can only be divided by itself and 1 only.
Examples of prime numbers are 2, 3, 5, 7, 11, 19 etc.
b
.
a
Factors and Multiples
When the number 12 is divided by any one of the numbers 1, 2, 3, 4, 6 and 12, the remainder
is zero. We call the numbers 1, 2, 3, 4, 6 and 12 factors of 12 and 12 is said to be a multiple
of 1, 2, 3, 4, 6 and 12. The number 12 is also said to be divisible by 1, 2, 3, 4, 6 and 12.
Prime Factors
The number 12 can be written as a product of its prime factors. i.e 12 = 22 x 3. This is called
‘Prime Factorization’. How do you write the number 60 as a product of its prime factors?
Example:
Written as the product of its prime factors, 360 = 23 x 32 x 5. Find the smallest integer k such
that 360k is a perfect cube.
To make 360k a perfect cube, we need to multiple (23 x 32 x 5) by (3 x 52) so that
360k = 23 x 33 x 53. Hence, k = 3 x 52 = 75.
It is also good for you to find out what are perfect squares and perfect cubes.
Primes and Prime Factorization
Prime Numbers are numbers that can be divided by itself and 1 only. For example, the
number 5 can only be divided by 5 and 1. It can’t be divided by any other number to give
zero remainder.
The smallest prime number is 2. It is the only even prime number. The numbers ‘0’ and ‘1’
1
are not prime numbers.
The list of prime number will hence be 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31…
The list is never ending. Notice that beside the number 2, the rest are all odd numbers. But
not all odd numbers are prime numbers. For example, the number 9 is not a prime number.
Prime Factorization
Prime factorization is a dividing process where a number keeps dividing by prime numbers
until it becomes 1. Usually, the division starts with the smallest possible prime number, i.e 2
or 3.
Example: Prime factorize the number 420.
2
2
3
5
7
420
210
105
35
7
1
Hence, the number 420 = 2 x 2 x 3 x 5 x 7
(product of its prime numbers)
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that
divides evenly into both numbers.
In other words the H.C.F is the largest of all the common factors.
The common factors or of 12 and 18 are 1, 2, 3 and 6.
The largest common factor is 6, so this is the H.C.F. of 12 and 18.
It is very easy to find a H.C.F. of small numbers, like 6 and 9 (it is 3) or 8 and 4 (it is 4).
The best way is to keep finding the factors of the smaller number, starting from the largest
factor. The first factor of the smaller number that is also a factor of the larger number is a
H.C.F.
For larger numbers you can use the following method:
1. Prime factorize both numbers.
2. Keep dividing until both numbers cannot be divided by the same prime number
anymore.
3. Multiply the numbers on the left column to get the HCF
Example:
Find the Highest Common Factor (H.C.F.) of 240 and 924.
Solution:
Prime Factorize 240 and 924 until:
2
2
2
3
240
120
60
20
924
462
231
77
Multiply the factors on the left column to get the H.C.F.
The Highest Common Factor is 2 x 2 x 3 = 12
------------------------------------------------------------------------------------------------------The Lowest Common Multiple (L.C.M) is the smallest number that is a common
multiple of two or more numbers.
For example, the L.C.M of 3 and 5 is 15.
The simple method of finding the L.C.M of smaller numbers is to write down the multiples of
the larger number until one of them is also a multiple of the smaller number.
Example :
Find the Lowest Common Multiple of 8 and 12.
Solution: Multiples of 12 are 12, 24...
24 is also a multiple of 8, so the L.C.M of 8 and 12 is 24.
To Find LCM of Large Numbers, we can also use prime factorization. But in this case,
multiply by the left column numbers as well as the bottom row numbers.
The LCM of 240 and 924 is therefore 2 x 2 x 3 x 20 x 77 = 18,480
Finding Squares, Cubes, Square Roots and Cube Roots by Prime Factorization
The square of a number is simply the number multiply by itself again. Example, the square of
3 is 9, i.e 3 x 3 = 9. We can write 3 square as 32.
A perfect square is a non-zero whole number that is produced by multiplying a whole number
by itself. Examples of perfect squares are: 1, 4, 9, 16, 25, 36, 49 …
The cube of a number is the multiplication of the number by itself three times. Example, the
cube of 3 is 3 x 3 x 3 = 27. We can write 3 cube as 33.
The square root of a positive number is the number when multiplied by itself, produce the
given number. The notation for square root is
. For example, the square root of 25 is 5.
We write as
25 5 .
The cube root of a given number is the number when multiplied by itself twice, produce the
given number. The notation for cube root is 3 . For example, the cube root of 64 is 4. We
3
write as 3 64 4 .
How to use prime factorization to find square root of a number? We see an example below.
Example:
Find the square root of 6561 using prime factorization.
3 6561
3 2187
3 729
3 243
3 81
3 27
3 9
3 3
1
Notice that there are altogether eight ‘3’s. Take four of the 3s to obtain 3 x 3 x 3 x 3 = 81.
Hence, the square root of 6561 is 81. How about cube root of 1331?
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Chapter 2: Fractions, Decimals and Percentages
Chapter 3: The Four Operations
Chapter 4: Negative Numbers
Chapter 5: Powers and Roots
Chapter 6: Ordering and Set Notations
Chapter 7: Ratio, Proportion and Rate
Chapter 8: Estimation and Limits of Accuracy
Chapter 9: Standard Forms and Surds
Chapter 10: Applying Numbers and using Calculators
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