NUMBER THEORY
TYPES OF NUMBERS
1)
EVEN NUMBERS: This is any whole number that is
exactly divisible by 2. For example { 0,2,4,6,8,…}
2)
ODD NUMBERS: This is any whole number that is
not even. This is any number that is not exactly divisible by 2. For example
{1,3,5,7,9,…}
3)
A PERFECT SQUARE NUMBER: This is a whole number which
is obtained when an integer is multiplied by itself.It means that when you
multiply an even or odd number by itself, you obtain a square number.
-From the even numbers above, 0×0=0, 2×2=4,
4×4=16, 6×6=36, 8×8=64,…
-From the odd numbers above, 1×1=1, 3×3=9,
5×5=25, 7×7=49,…
A set of perfect square numbers is select
from above as follows: {0, 1,4,9,16,25,49,64,…}
4)
FACTORS: The factor of a whole number is a whole
number that divides exactly into it to give another whole number. For example 4
is a factor of 12 because it divides 12 into 3 equal parts. That is 12÷4=3.
Another name for factors is divisors. A single number may have several factors (divisors).
From above the divisor 4 , and the quotient
3 are both factors of 12. Recall that 3 and 4 are factors of 12 because 3×4=12.
Also, 1×12=12, means 1 and 12 are factors of 12.Hence 12 is a factor of itself just
like every other number While 1, is the
factor of itself and every other number. The factors are often written as a set
and denoted by D(). The factors of 12 is denoted by D(12)
HOW TO FIND THE FACTORS OF A NUMBER
To find the factors of a number, we try
dividing the number by 1,2,3,4,…etc in turn. An exact division tells you about
two factors.
Example 1. Write down the factors (or
divisors ) of 6.
Solution
Divide 6 by 1,2,…,6 and write the result
as a set denoted by D(6).
6÷1=6
6÷2=3
6÷3=2
6÷4=1R2
6÷5=1R1
6÷6=1
All answers without remainders are factors
of 6 because they divide exactly into it.
D(6)={1,2,3,6)
Example 2: Write down all the factors of
16.
Solution
16÷1=16
16÷2=8
16÷3=4R4
16÷4=4
16÷5=3R1
16÷6=2R4
16÷7=2R2
16÷8=2
16÷9=1R7
16÷10=1R6
16÷11=1R5
16÷12=1R4
16÷13=1R3
16÷14=1R2
16÷15=1R1
16÷16=1
All answers without remainders are the
factors of 16.
D(16)={1,2,4,8,16}
To solve faster, always ignore results that
have remainders as these are usually not factors. Go straight away to results
that do not have remainders.
Example 3. Find the factors of 5.
Solution
5÷1=5Î
5÷2=2R1
5÷3=1R2
5÷4=1R1
5÷5=1
All answers without remainders are factors
of 5
D(5)= {1,5}.
The number 5 has just two factors, 1, 5 itself.
Any number that has only two factors like 5 is called a prime number.
COMMON FACTORS AND
HIGHEST COMMON FACTORS
a)COMMON FACTORS. They are also known as
common divisors. To find the common factors of numbers say 12 and 18, simply
look for D(12)Î D(18).
Example 1. Find the common
factors of 8 and 12.
Solution
D(8)={1,2,4,8}
D(12)={1,2,3,4,6,12}
Common factors ;D(8)Î D(12)={1,2,4}
b) HIGHEST COMMON FACTORS. Among the factors
in D(8)Î D(12) ={1,2,4}, the highest is 4.We say the Highest Common Factor (HCF) of
8 and 12 is 4. Otherwise written HCF(8,12)=4.
We can also say the Greatest Common Divisor
of 8and 12 is 4 or GCD(8,12)=4.
Example 2. i) write down
common factors of 30 and 32.
ii) Find
HCF(30,32)
iii) Find GCD(30,45),
SOLUTION
i-
D(30) ={1,2,3,5,6,10,15,30} ,
D(32)={1,2,4,8,16,32} , D(30)Î D(32)={1,2}
ii-
HCF(30,32)=2
iii-
D(30)={1,2,3,5,6,10,15,30} ,D(45)={1,3,5,15,45},
D(30)Î D(45)={1,3,5,15}
GCD(30,45)=15
MULTIPLES AND LOWEST COMMON MULTIPLES
The numbers 3 and 4 are factors of 12.So 3×4=12. We say 12 is the
multiple of 3 and 4. There are other multiples of 3 and 4. The multiples of 3
is dented by M(3). Same with the multiples of 4 ie M(4).
Generally a number P is a multiple
of another number Q if Q is the factor of P.
For example
8 is a multiple of 2 because 2 is a factor of 8 ie 2×4=8
9 is a multiple of 3 because 3 is a factor of 9 ie 3×3=9
7 is a multiple of 1 because 1 is a factor of 7 ie 1×7=7.
Since 1 is the factor of every number, every number is a multiple of 1.
Zero(0) is a trivial multiple of every number because the product of any number
and zero gives zero. In most cases,however,0 is usually neglected.
Example1. Write out members of M(3)
and M(2) for numbers less than
20.
Solution
M(3)= {3,6,9,12,15,18}
M(2)={2,4,6,8,10,12,14,16,18}
COMMON MULTIPLES
From above, the common multiple is found by considering M(3)Î M(2). From
this,
M(3)Î M(2)={ 6,12,18}
LEAST COMMON MULTIPLES (LCM)
The least common multiple of two numbers x and y is denoted by
LCM(x,y) .It is the smallest number in
M(x)Î M(y).
From M(3)Î M(2)={6,12,18} , the
least of them is 6; written LCM(3,2)=6.
Example 2: Find the LCM of 8 and 10 for numbers less than 70
Solution
M(8)= {8,16,24,32,40,48,56,64}
M(12)={12,24,26,48,60}
M(8)Î M(10)={24,48}
LCM(8,12)=24.
We shall study a proper way to calculate LCM and HCF after
studying prime numbers.
5)
PRIME NUMBERS :Consider the following
numbers 1,2,3,4,5.Their factors are
written as follows:
D(1)={1}
D{2}={1,2}
D{3}={1,3}
D(4)={1,2,4}
D(5)={1,5}
D(7)={1,7}
Observe that 2,3, 5,and 7 have only two
factors. 4 has more than two factors while 1 has only one factor. We say 2,3, 5
and 7 are prime numbers because each has two factors.
DEFINITION: A prime number is a number
which has exactly two factors. Example {2,3,5,7,11,13,17,…}.
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