Basic Mathematics Form One Notes – Factors, Multiples and Integers
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8. Hamis uses 200 shillings every day for transport. How much money will he use for 35 days?
Factors And Multiples Of Numbers
Factors of a Number
Find factors of a number
Consider two numbers 5 and 6, when we multiply these numbers ie. 5 6 the answer is 30, The
numbers § and 6 are called factors or divisors of 30 and number 30 is called a multiple of S and
6. Therefore factors are the divisors of a number
Example 15
Find all factors of 12
Note that, when listing the factors we don’t repeat any of it
Write 12 as a product of two numbers
12=1×12
=2×6
=3×4
=4×3
=6×2
=12×1
Therefore, the factors of 12 are 1, 2, 3, 4, 6, 12.
Note that, when listing the factors we don’t repeat any of it.
Multiples of a Number
Find multiples of a number
Multiples of a number are the products of a number. For example multiples of 4 are 4, 8, 12, 16,
20, 24, … This means multiply 4 by 1, 2,3, 4,5, 6,
Example 16
Example
List all multiples of 6 between 30 and 45
16
Solution
The multiples of 6 are: 6*1 = 6; 62 = 12; 63 = 18; 64 = 24; 6S = 30; 66 = 36; 6*7 = 42;
68 = 48 and so on
Therefore, multiples of 6 between 30 and 45 are 36 and 42
Factors to Find the Greatest Common Factors(GCF) of Numbers
Use factors to find the greatest common factors(GCF) of numbers
Greatest Common Factor is sometimes called Highest Common Factor. Its short form is (GCF)
or (HCF) respectively. The Greatest Common Factor is the largest common divisor of two or
more numbers given.
For example if you are told to find the GCF of 15 and 25. First, list all factors or divisors of 15
and that of 25. Thus, factors of 15 are 1, 3, 5, 1Sfactors of 25 are 1, 5, 25
‘The common factors are 1 and 5. Therefore the GCF is 5
Example 17
Example!
Find the HCF of 72 and 120.
Solution
We have to list factors of our numbers
Factors of 72 are 1, 2,3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 120 are 1, 2, 3, 4, 5, 6,8, 10, 12, 15, 20, 24, 30, 40, 60, 120
The common factors are 1, 2, 3, 4, 6, 8, 12, 24.
Therefore the HCF of 72 and 120 is 24.
‘Another method which can be used to find the GCF or HCF is prime factorization method
Example 18
v
Example
Find the HCF of 36 and 48,
Solution
We have to find prime factors of 36 and 48 first
2) 36 2248
2)18 2)24
3)9 2)12
3)3 2) 6
1 3).3
1
thus, 36 = 2«23%3
48 = 2x2x2* 2×3.
After writing the numbers as a product of their prime factors, take only the common prime
factors (prime factors appeared to all numbers). In our example the common factors are 223
Therefore the HCF of 36 and 48 is 12.
Lowest Common Multiple (LCM)
Lowest Common Multiple is also called Least Common Multiple and its short form is LCM.For
example: multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, … multiples of 3 are 3, 9, 15,
18, 21, 24, 27, 30, 33,
When you look carefully at these multiples of 5 and 3 you notes that 15 had appeared to both.
This multiple which appear to both is called a common multiple. If there are more than one
common multiples which appeared the smaller common multiple is what is called Lowest
Common Multiple.
Example 19
18
Find the common multiples and then show the Lowest Common Multiple of the numbers 4, 6
and 8.
Solution
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56,
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72.
The common multiples of 4, 6 and 8 are 24, 48,
Therefore the Lowest Common Multiple of 4, 6, and 8 is 24.
We can find the Lowest Common Multiple (LCM) of numbers by writing the numbers as a
product of their prime factors. The method is called prime factorization
Example 20
Find the LCM of 24 and 36 by prime factorization method.
Solution
Let us find prime factors of each number by dividing the numbers by their prime factors.
2)24 2)36
2)12 2)18
2) 6 3)_9
3.3 33
1 1
Thus, 24=2x2x2x3.
36=2x2x3x3.
Now take the prime factors which appear to both numbers ie. 2×23 (we take without
repeating). We are left with 2 which is a multiple of 24 and 3 which is a multiple 36. We have
also to multiply these prime factors left ie. 2×2*2*3×3. This gives 72. Therefore the LCM of 24
and 36 is 22*2x3x3= 72,
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Integers
Integers
Identify integers
Consider a number line below
a
5S 4 3 2 1000 1 2 3 4 5
The numbers from 0 to the right are called positive numbers and the numbers from 0 to the left
with minus (-) sign are called negative numbers. Therefore all numbers with positive (+) or
negative (-) sign are called integers and they are denoted by Z. Numbers with positive sign are
written without showing the positive sign. For example +1, +2, +3, … they are written simply as
1,2, 3, … . But negative numbers must carry negative sign (-). Therefore integers are all positive
and negative numbers including zero (0). Zero is neither positive nor negative number. It is
neutral
The numbers from zero to the right increases their values as the increase. While the numbers
from zero to the left decrease their values as they increase. Consider a number line below.
a a er ad
S 4 3 2 a o 1 2 3 4 5
If you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater than 2. We use the symbol
*> to show that the number is greater than i, e. 3 >2(three is greater than two). And since 2 is to
the left of 3, we say that 2 is smaller than 3 ie. 2<3 =”” div=” is=TM” less=”” number=””
show=”” symbol=”” than.=”” that=”” the=”” to=”” use=”>
Consider numbers to the left of 0. For example if you take -5 and -3. -5 is to the left of -3,
therefore -5 is smaller than -3.-3 is to the right of -5, therefore -3 is greater than -S.
Generally, the number which is to the right of the other number is greater than the number which
is to the left of it
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If two numbers are not equal to each to each other, we use the symbol “¥° to show that the two
numbers are not equal. The not equal to “2 is the opposite of equal to ‘=”
Example 21
Represent the following integers Z on a number line
a is greater than Z and Z is greater than -4
b. -2is ess than Z and Z.is less than or equal to 1
Solution
a. 0 is greater than Z means the integers to the left of zero and Z is greater than -4 means integers
to the left of 4. These numbers are -1, -2 and -3. Consider number line below
<<.
3 2 a
b. -2 is less than Z means integers to the right of -2 and Z is less than or equal 1o 1 means
integers to the left of | including 1. These integers are -1, 0. and 1, Consider the number line
below
—_ _,_ ~~
a o 1
Example 22
Put the signs “is greater than’ (>), ‘is less than’ (<), ‘is equal to” (=) to make a true statement.
a4 []-s 4. 78 [] 2
b. 7 | Zz e. -25 [] -37
« 32 | | 3 f. 20 [Jos
a
Solution
a. -4 5 d. 78 [= | 78
»7 [2] 2 e. -25 37
c. -32 -3 f. 20 > 5
Addition of Integers
Add integers
Example 23
2+3
Show a picture of 2 and 3 on a number line.
Solution
Show a picture of 2 and 3 on a number line.
—
or re
a0 49 2 | 3
p24 ee
Then add 2 and 3. It will appear as here below
—_—
a0 1 2 3 4 5
2 3
Re}
2+3=5
When drawing integers on a number line, the arrows for the positive numbers goes to the right
while the arrows for the negative numbers goes to the left. Consider an illustration bellow.
2
a
3 2 1 o 1 2 3
‘ t———>|
— 3
The distance from 0 to 3 is the same as the distance from 0 to -3, only the directions of their
arrows differ. The arrow for positive 3 goes to the right while the arrow for the negative 3 goes
to the left
Example 24
3+6
Solution
OT To naw >
3 2-1 o 1 2 3 4 5
34 6
-—3—_+
Subtraction of Integers
‘Subtract integers
Since subtraction is the opposite of addition, if for example you are given 5-4 is the same as 5 +
(-4). So if we have to subtract 4 from 5 we can use a number line in the same way as we did in
addition. Therefore $4 on a number line willbe
—_ —_——_— >
90 1 2 3 4 5
5 4
t >
23
Take five steps from 0 to the right and then four steps to the left fom S. The result is 1
Multiplication of Integers
Multiply integers
Example 25
2×6 is the same as add 2 six times ie. 26=2+2+2+2+2+2= 12. On anumber line will be:
~S 2 2 2 2 2
o 1 2 3 4 5 6 7 8 $8 10 1 2
Example 2
“3×4
Solution
“3×45 -34-34-34-3=-12
On a number line will be
3 3 3 3
en en a
V \
31241 109 8 7 6 5 4 3 2 1 0
Multiplication of a negative integer by a negative integer cannot be shown on a number line but
the product of these two negative integers is a positive integer.
From the above examples we note that multiplication of two positive integers is a positive
integer. And multiplication of a positive integer by a negative integer is a negative integer. In
summary:
5 ORHEH)
+ OSO=H)
+ OOFO)
+ OOO)
24
Division of Integers
Divide integers
Example 26
63 is the same as saying that, which number when you multiply it by 3 you will get 6, that
number is 2, so, 6=3 = 2.
Therefore division is the opposite of multiplication. From our example 23 = 6 and 6+3 = 2
Thus multiplication and division are opposite to each other.
Dividing two integers which are both positive the quotient (answer) is a positive integer. If they
are both negative also the quotient is positive. If one of the integer is positive and the other is
negative then the quotient is negative. In summary
5 OOO)
+ OR)
+ FORO)
+ FARO)
Mixed Operations on Integers
Perform mixed operations on integers
‘You may be given more than one operation on the same problem. Do multiplication and division
first and then the rest of the signs. If there are brackets, we first open the brackets and then we do
division followed by multiplication, addition and lastly subtraction. In short we call it BODMAS.
The same as the one we did on operations on whole numbers,
Example 27
93 +302-1=
Solution
953430241
=3 + 6-1 first divide and multiply)
23
=8 (add and then subtract)
Example 28
(12+4-2)+4-7=
Solution
(12+4-2)+4-7
=1 + 4—7 (do operations inside the brackets and divide first)
=5~7 (add)
2
FRACTIONS
AA fraction is a number which is expressed in the form of a/b where a – is the top number called
numerator and b- is the bottom number called denominator.
26