1. BASIC ARITHMETIC OPERATIONS • Differentiate between a factor and multiple • Define a prime ...

117 Pages·2017·1.55 MB·English

1. BASIC ARITHMETIC OPERATIONS

• Differentiate between a factor and multiple
• Define a prime factor
• Prime Factorization/ Decomposing into Prime Factors
• Define Highest Common factor (HCF) and Lowest Common Multiple
(LCM)
• Determine HCF and LCM
• Applications of HCF and LCM
Factors
The factors of a number are those numbers which divide exactly into the
number.
Factor – a number that is multiplied by another to give a product.
e.g. The factors of 24 are 1, 2, 3, 4, 6, 12, 24
The pairs of factors of 30 are 1 x 30, 2 x 15, 3 x 10 and 5 x 6 therefore the
factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30
Multiples
n kn k
A multiple of a number is where is a counting number.
Multiple – An answer to a multiplication problem, e.g 7856, 56 is a
multiple
Examples:
Some multiples of 5 are 5,10,15, …
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, …
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Prime Numbers
A natural number x1 is said to be prime if and only if it is divisible by 1 and
itself. Meaning, a prime number has only two natural factors, itself and 1. Zero
and 1 are not prime numbers. Otherwise, a number which is not a prime is
called a composite - i.e. it is composed of more than two natural factors.
Examples
 Primes: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,etc.
 Composites: 4,6,8,9,10,12,14,15,16,18,20, 21,22,24,etc.
Example: 7 is prime because the only numbers that will divide into it evenly are
1 and 7.
Few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 etc.
Prime Factors
Remember the factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 , The prime factors of
30 are 2, 3, and 5
We can find the prime factors by expressing a number as a product of a
primes.
How to express a number as a product of primes or product of prime factors:
E.G. Express 360 as a product of primes
1. Write down the 1st few prime numbers e.g. 2,3,5,7,11,13
2. Divide 360 by the 1st prime number (2) as many times as possible
until it can no longer divide exactly into that number.
3. Divide 360 by the next prime number (3) as many times as
possible and so on until you get 1.
4. Write down the product of all the prime numbers you divided in.
One way of doing this is:
360 180 90 45 15 5
: : : : : :1 This means that 360222335
2 2 2 3 3 5
5. We can write any repeated prime number as powers (using index
form) as: 36023325

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Exercise
Express 378 and 18522 as a product of prime numbers.

Highest Common Factors ( HCF) and Highest Common Divisor (HCD)
To find the HCF:
Method 1: List all factors of the given numbers and identify the HCF.
For example, to the highest common factor of 8 and 12
Factors of 12 are: 1, 2, 3, 4, 6, 12
Factors of 8 are: 1, 2, 4, 8
Those bolded factors are common factors but the biggest common factor is 4.
Hence, the HCF is 4.
Method 2: Factorise the given numbers into their prime factors respectively.
Select those numbers (with lowest power) which occur commonly in all of the
decompositions and multiply them together.
12 6 3 2
For twelve : : :1122 3
2 2 3
8 4 2 3
For eight : : :182 1
2 2 2
2
HCF 2 4
Exercise
Find the HCF of:
(a) 24, 72, 96 and 300
(b) 255 and 75

Lowest Common Multiple (LCM)
Method 1: List the multiples of the given numbers and identify the LCM.
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Find the LCM of 12 and 8
Multiples of 8 are 8, 16, 24, 32, 40,...............
12 = 12, 24, 36, 48,.............
We see that the very first (lowest) number appears in both lists is 24, hence we
say that 24 is the LCM of 12 and 8.
Method 2: Factorise the given numbers into their prime factors respectively.
Select every number (prime factors) with its highest power which occur in any
of the decompositions (prime factors) of each of the given numbers and
multiply them together.
To find the LCM of 12 and 8, we decompose as follows:
12 6 3 2
For 12 we have : : :1whichis 122 3
2 2 3
8 4 2 3
For 8 we have : : :1whichgives82
2 2 2
3
Hence, LCM = 2 324
Exercise:
Find the LCM of
(a) 72, 96 and 300
(b) 455, 1050

Problem sums on HCF and LCM can be really tricky as they are not easy to
identify. Thus in this case the main focus is not on going through how to find
HCF and LCM (please refer to your notes on those), but more importantly to go
through how to determine when to find the HCF and when to find the LCM of
the numbers involved in the problem sums.

Let’s take a look at a typical problem involving the HCF.

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HCF – Example

3 strings of different lengths, 240 cm, 318 cm and 426 cm are to be cut into
equal lengths. What is the greatest possible length of each piece?
If you notice, finding the HCF is crucial here because you are trying to find what
the 3 numbers have in common, i.e. a common factor. All 3 numbers must be
able to be divided by the same number in order for all 3 strings to be cut into
equal lengths. HCF is needed here because you are asked to find the greatest
possible length.
Therefore,

LCM – Example
Two lighthouses flash their lights every 20s and 30s respectively. Given that
they flashed together at 7pm, when will they next flash together?
One method to finding the next time the lighthouses flash together is to list
the seconds:
20, 40, 60
30, 60, 90
60 is a multiple common to 20 and 30, and thus the lighthouses will flash
together in 60s’ time, i.e. at 7:01pm.
This is the same as finding the lowest common multiple, or LCM:
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There are other different types of problems involving LCM, but
just remember that such questions involve you trying to find a
multiple that is common to the numbers involved.

1. As a humanitarian effort, food ration is distributed to each
refugee in a refugee camp. If a day’s ration is 284 packets of
biscuits, 426 packets of instant noodles and 710 bottles of
water, how many refugees are there in the camp? [142
refugees]

2. 294 blue balls, 252 pink balls and 210 yellow balls are
distributed equally among some students with none left
over. What is the biggest possible number of students? [42
students]
3. A group of girls bought 72 rainbow hairbands, 144 brown
and black hairbands, and 216 bright-coloured hairbands.
What is the largest possible number of girls in the group?
[72 girls]

4. A man has a garden measuring 84 m by 56 m. He wants to
divide them equally into the minimum number of square
plots. What is the length of each square plot? [28 m]

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5. Leonard wants to cut identical square as big as he can from
a piece of paper 168 mm by 196 mm. What is the length of
each square? [28 cm]

6. A small bus interchange has 2 feeder services that start
simultaneously at 9am. Bus number 801 leaves the
interchange at 15-min intervals, while bus number 802
leaves at 20-min intervals. On a particular day, how many
times did both services leave together from 9 am to 12 noon
inclusive? [4 times]

7. Candice, Gerald and Johnny were jumping up a flight of
stairs. Candice did 2 steps at a time, Gerald 3 steps at time
while Johnny 4 steps at a time. If they started on the bottom
step at the same, on which step will all 3 land together the
first time? [12th step]

8. Heidi helps out at her mum’s stall every 9 days while her
sister every 3 days. When will they be together if they last
helped out on June 16 2008? [June 25 2008]

9. A group of students can be further separated into groups of
5, 13 and 17. What is the smallest possible total number of
students? [1105 students]

10. Jesslyn goes to the market every 64 days. Christine goes to
the same market every 72 days. They met each other one
day. How many days later will they meet each other again?
[576 days]

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FUNDAMENTAL OPERATIONS ON WHOLE NUMBERS

Directed Numbers
To add two directed numbers with the same sign, find the sum of the numbers
and give the answer the same sign.
Examples
3(5)  35 8
7(3)  73 10

9.1(3.1) 9.13.1 12.2
2(1)(5) (21)5 35 8
To add two directed numbers with different signs, find the difference between
the numbers and give the answer the sign of the larger number.
Examples
7(3)  73 4
9(12)  9123
8(4)  84 4
To subtract a directed number, change its sign and add.
Examples
7(5)  75 2
7(5)  75 12

8(4)  84 12
9(11)  911 2
Rules for multiplications
Pos. number x pos. number = pos. number
Neg. number x neg. number = pos number
Neg. number x pos. number = neg. number
Pos. number x neg. number = neg. Number
Same goes for division.
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BASIC ARITHMETIC
Rules of Arithmetic
BEDMAS (brackets, exponents, division, multiplication, addition and
subtraction)
BODMAS (brackets, powers, division, multiplication, addition and
subtraction)
1. Work out brackets
2. work out powers (exponents)
3. Divide and multiply
4. Add and subtract
2  2
1. 3 3 32 7411
2. 242384(7)
2
3. 24 2214
Simplify each of the following
1. 23725623
   
2. 42186441210
 
5 2
3. 2 327(8)(5 6)44

4. 5347669
 
5. 81320954
6. 722(24)2424(35)17
   

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Vulgar Fractions – Concepts and operations
7 7
In a fraction 7 is a numerator and 8 is a denominator. is referred
8 8
8
to as a proper fraction and is referred to as a improper.
7
7
2 is referred to as a mixed number.
8
Equivalent fractions:
2 4 6 8
  
3 6 9 12

OPERATIONS
Evaluate and simplify your answer.

3 4 1 2 3 4 1 2
1.  2.  3.  4. 
9 9 4 3 9 9 4 3

1 2 1 2 1 2 2 1
5.  6.  7. 1 2 8. 2 
4 3 4 3 3 5 3 4
2 1

3 5 3 1 7 2 1 2  1
9. 10.    11. 1 2  41
3 1 8 5 10 3 2 3  3
  
4 3

2 1 7 2 3 2 3 3 7 5
12. 2 1 53 13.    14. 2 2  
   
3 2 8 7 5 3 7 4 8 8
   
Fractions Word Problems
1. Johannes had 640 shares. He sold out one third of them to a trading
2
company and of the remainder to another company. How many
5
shares remained with Johannes?
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