JSS 2 Mathematics First Term Week 1

Topic: WHOLE NUMBERS

Factors, Prime factors (revision)

40 ÷ 8 = 5 and 40 ÷ 5 = 8

8 and 5 divide into 40 without remainder.

8 and 5 are factors of 40.

A prime number has only two factors, itself and 1, 2, 3, 5, 7, 11, 13, … are prime numbers.

1 is not a prime number.

Common factors

The number 12, 21 and 33 are all divisible by 3. We say that 3 is a common factor 0f 12, 21 33.

There may be more than one common factor of a set of numbers. For example, both 2 and 7 are common factors of 28, 42 and 70. Since 2 and 7 are common factors and are both prime numbers, then 14 (= 2 x 7) must also be a common factors of the set of numbers.

1 is a common factor of all numbers.

Highest Common Factor (HCF)

2, 7 and 14 are common factors of 28, 42 and 70; 14 is the greatest of three common factors. We say that 14 is the highest common factor of 28, 42 and 70.

To find the HCF of a set of numbers:

Express the number as a product of prime factors;

b. Find the common prime factors

c. Multiply the current prime factor together to give the HCF.

Example

Find the HCF of 18, 24 and 42.

18 = 2 x 3 x 3

24 = 2 x 2 x 2 3

42 = 2 x 3 x 7

The common prime factors are 2 and 3.

The HCF = 2 x 3 = 6.

Find the HCF of 216 and 288

2 | 216

2 | 108

2 | 54

3 | 27

3 | 9

3 | 3

……..

0      1

2 | 144

2 | 72

2 | 36

2 | 18

3 |   9

3 |    3

……..

0      1

In index notation

216 = 23 x 33

288 = 25 x 32

23 is the lowest power of two contained in the two numbers. Thus the HCF contains 23.

32 is the lowest power of 3 contained in the tow numbers. The HCF contains 32.

216 = (23 x 33) x 3

288 = (22 x 33) x 22

The HCF = 22 x 33 = 8 x 6 = 72

Rules of divisibility

Table 1.2 gives some rules for divisors of whole numbers.

Any whole number is exactly divisible by …
2 if its last digit is even or 0
3 if the sum of its digit is divisible by 3
4 if its last two digits form a number divisible by 4
5 if its last digit is five or zero
6if its last digit is even and the sum of its digits is divisible by 3
8 if its lat three digits forma number divisible by 8
9if the sum of its digit is divisible by 9
10 if its last digit is 0

Table

There is no easy rule for division by 7.

Notice the following:

a. If a number m is divisible by another number n, m is also divisible by the factors of n. For example, a number divisible by 8 is also divisible by 2 and 4.

b. If a number is divisible by two or more numbers, it is also divisible by the LCM of these numbers. For example, a number divisible by both 6 and 9 is also divisible by 18, 18 is the LCM of 6 and 9.

Example

Test the following numbers to see which are exactly divisible by 9. a. 51 066 b. 9 039

Solution

a. 5 + 1 + 0 + 6 + 6 = 18

18 is divisible by 9.

Thus 51 066 is divisible by 9.

b. 9 + 0 + 3 + 9 = 21

21 is not divisible by 9.

Thus 9 039 is not divisible by 9.

Number patterns

The multiples of 3 can be given in a row, or sequence: 3, 6, 9, 12, 15, 18, 21, …

They can also be shown by thickening on a number 1 – 100 number square as in fig below

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 10
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

These are both examples of number patterns.

Find the next four terms of the following sequences.

a. 2, 5, 8, 11, 14, _, _, _, _

b. 1, 6, 11, 16, 21, _, _, _, _

c. 1, 12, 23, 34, 45, _, _, _, _

Graphs and number patterns

A graph is a picture. The pictograms, bar charts, and pie charts are all examples of graphs.

Graphs are usually drawn on graph paper. There are two common kinds of graph paper as shown below.

The lines on the graph paper are thick, medium or thin. This make big, medium and small squares. On your graph paper, find out the following

 

2mm graph paper; the small squares are 2mm by 2mm

1mm graph paper; the small squares are 1mm by 1mm

1. the length of side of the big medium, and small squares;

2. the number of small squares inside a big square;

3. the width in big squares of your graph paper;

4. the length, in big squares, of your graph paper.

Squares and square roots

Square roots

72 = 7 x 7 = 49.

In words ‘ the square of 7 is 49’. We can turn this statement round and say. ‘the square root of 49 is 7’.

In symbols, √49 = 7. The symbol √ means the square root of .

To find the square root of a number, first find its factors.

Example

Find √11 025.

Method: Try the prime numbers 2, 3, 5. 7, …

Working:

3 11 025
3  3 675
5  1 225
5    245
7      49
7       7
        1

11 025 = 32 x 52 x 72

= (3 x 5 x7) x (3 x 5 x 7)

= 105 x 105

Thus √11 025 = 105

It is not always necessary to write a number in its prime factors.

Example

√6 400

6400 = 64 x 100

= 82 x 102

Thus √6 400 = 8 x 10 = 80

The rules for divisibility can be useful when finding square root.

Exercise

Find by factors the square roots of the following:

1. 225

2. 194

3. 342

4. 484