01_chapter01_001-019

Using exactly four fours, and usual mathematical symbols, try to make eachwhole number from 1 to 100. Here are a few examples to start you off.

1 � �4

4

4

4�

2 � �4

4

�

�

4

4�

3 � �4 � 4

4

� 4�

4 � 4 � 4 � (4 � 4)

5 � �4 � 4

4

� 4�

6 � 4 � �4 �

4

4�

You should try to stick to basic mathematical symbols such as �, �, �, � andbrackets, wherever possible, but you may need to use more complicatedsymbols such as �� and ! to make some of the higher numbers. Ask your teacherif you need some help with these symbols.

1 Working with whole numbers 1

CHAPTER 1

Working with whole numbers

In this chapter you will revise earlier work on:

• addition and subtraction without a calculator• multiplication and division without a calculator• using positive and negative whole numbers (integers)• factors and multiples.

You will learn how to:

• decompose integers into prime factors• calculate Highest Common Factors (HCFs) and Lowest Common

Multiples (LCMs) efficiently.

You will also be challenged to:

• investigate primes.

Starter: Four fours

44

44

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1.1 Addition and subtraction without a calculator

You will sometimes need to carry out simple addition and subtraction problems inyour head, without a calculator. These examples show you some useful shortcuts.

EXAMPLE

Work out the value of 19 � 6 � 21 � 4.

SOLUTION

19 � 6 � 21 � 4 � 19 � … � 21� 6 … � 4

� 40 � 10� 50

EXAMPLE

Work out the value of 199 � 399.

SOLUTION

199 � 399 � 200 � 1 � 400 � 1� 200 … � 400

� 1 … � 1� 600 � 2� 598

EXAMPLE

Work out 257 � 98.

SOLUTION

257 � 98 � 257 � 100 � 2� 157 � 2� 159

Harder questions may require the use of pencil and paper methods, and you shouldalready be familiar with these. Remember to make sure that the columns are linedup properly so that each figure takes its correct place value in the calculation.

2 1 Working with whole numbers

When adding a string of numbers, look for combinationsthat add together to give a simple answer. Here, 19 � 21and 6 � 4 both give exact multiples of 10.

Both these numbers are close to exact multiples of 100,so you can work out 200 � 400 and then make a smalladjustment.

98 is close to 100, so it is convenient totake away 100, then add 2 back on.

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EXAMPLE

Work out 356 � 173.

SOLUTION

So 356 � 173 � 529

Here are two slightly different ways of setting out a subtraction problem. You should use whichever of these methods you prefer.

EXAMPLE

Work out 827 � 653.

SOLUTION

Method 1

So 827 � 653 � 174

Method 2

So 827 � 653 � 174

18 2 7

�76 5 31 7 4

8 2 7� 6 5 3

4

7 18 2 7

� 6 5 31 7 4

8 2 7� 6 5 3

4

3 5 6� 1 7 3

5 2 91

3 5 6� 1 7 3

2 91


Work from right to left.

Add the units: 6 � 3 � 9

Next, the 10s column: 5 � 7 � 12

The digit 2 is entered, and the 1 is carried to the next column.

Finally, the 100s column: 3 � 1 � 1 � 5

For the units: 7 � 3 � 4

For the 10s: 2 � 5 cannot be done directly.

The first part is the same as method 1.

Now 12 � 5 � 7 and 8 � 7 � 1

Exchange 10 from the 82 to give 70 and 12.

Now 12 � 5 � 7 and 7 � 6 � 1

Instead of dropping 82 down to 72, you can make 65 up to 75.

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1.2 Multiplication without a calculator

You will sometimes need to carry out simple multiplication problems in yourhead. This example shows one useful shortcut.

EXAMPLE


SOLUTION

49 � 50 � 1

So 49 � 3 � 50 � 3 � 1 � 3� 150 � 3� 147

Harder questions will require pencil and paper methods. Here is a reminder ofhow short multiplication works.

EXAMPLE



49 is almost 50, so you can work out50 � 3 then take off the extra 3.

EXERCISE 1.1Work out the answers to these problems in your head.

1 46 � 19 � 54 � 11 2 198 � 357 � 2 3 66 � 111 � 14

4 345 � 187 � 55 5 23 � 24 � 25 � 26 � 27 6 39 � 48 � 61 � 52

7 59 � 69 � 79 8 144 � 99 9 149 � 249

10 376 � 199

Use any written method to work out the answers to these problems. Show your working clearly.

11 274 � 89 12 456 � 682 13 736 � 473

14 949 � 477 15 1377 � 2557 16 3052 � 1644

17 6355 � 2471 18 2005 � 1066

19 An aircraft can carry 223 passengers when all the seats are full, but today 57 of the seats are empty. How many passengers are on the aircraft today?

20 The attendances at a theatre show were 475 (Thursday), 677 (Friday) and 723 (Saturday). How many people attended in total?

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SOLUTION

So 273 � 6 � 1638

When working with bigger numbers, you will need to use long multiplication.There are two good ways of setting this out – use whichever one you are mostconfident with.

EXAMPLE


SOLUTION

Method 1

So 492 � 34 � 16 728

4 9 2� 3 41 9 6 8

1 4 7 6 01 6 7 2 8

4 9 2� 3 41 9 6 8

1 4 7 6 0

4 9 2� 3 41 9 6 8

0

4 9 2� 3 41 9 6 8

2 7 3� 6

1 6 3 81 4 1

2 7 3� 6

3 84 1

2 7 3� 6

81


Begin with 3 � 6 � 18. Enter as 8 with the 1 carried.

Next, 7 � 6 � 42, plus the 1 carried,makes 43. Enter as 3 with the 4 carried.

Finally, 2 � 6 � 12, plus the 4 carried,makes 16. Entered as 6 with the 1carried; enter this 1 directly into the1000s column.

First, multiply 492 by 4 to give 1968.

Next, prepare to multiply 492 by 30,by writing a zero in the unitscolumn. This guarantees that you aremultiplying by 30, not just 3.

492 times 3 gives 1476.

1 3

2

Finally, add 1968 and 14 760 to give the answer 16 728.1 1

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SOLUTION

Method 2


Finally, add up the totals along eachdiagonal, starting at the right and workingleftwards.

1 6 7 2 81 1

36

08

4 9 2

3

416

36

08

27

06

12

36

08

4 9 2

3

416

36

08

27

06

12

EXERCISE 1.2Use short multiplication to work out the answers to these calculations.

1 144 � 3 2 254 � 4 3 118 � 6

4 227 � 8 5 326 � 7 6 420 � 5

7 503 � 4 8 443 � 9

Use any written method to work out the answers to these problems. Show your working clearly.

9 426 � 12 10 255 � 27 11 308 � 21

12 420 � 49 13 866 � 79 14 635 � 42

15 196 � 88 16 623 � 65

Within each square of the grid, carryout a simple multiplication as shown.For example, 9 times 3 is 27

First, write 492 and 34 along the top anddown the end of a rectangular grid.

Next, add diagonal lines, as shown.

4 9 2

3

4

4 9 2

3

4

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1.3 Division without a calculator

Division is usually more awkward than multiplication, but this example showsa helpful method if the number you are dividing into (the dividend) is close to aconvenient multiple of the number you are dividing by (the divisor).

EXAMPLE


SOLUTION

693 is 700 � 7

So 693 � 7 � 700 � 7 � 7 � 7� 100 � 1� 99

In most division questions you will need to use a formal written method. Here is an example of short division, with a remainder.

EXAMPLE


SOLUTION

4��6�7�3�

14��6�27�3�

1 64��6�27�33�

1 6 84��6�27�33� remainder 1

So 673 � 4 � 168 r 1 (or 168�14�)


693 is almost 700, so you can work out700 � 7 then take off the extra 7 � 7

First, set the problem up using this division bracket notation.

Divide 4 into 6: it goes 1 time, with a remainder of 2.

Next, divide 4 into 27: it goes 6 times, with aremainder of 3.

17 A company has 23 coaches and each coach can carry 55 passengers. What is the total number of passengers that the coaches can carry?

18 I have a set of 12 encyclopaedias. Each one has 199 pages. How many pages are there in the whole set?

19 Joni buys 16 stamps at 19 pence each and 13 stamps at 26 pence each. How much does she spend in total?

20 A small camera phone has a rectangular chip of pixels that collect and form the image. The chip size is 320 pixels long and 240 pixels across. Calculate the total number of pixels on the chip.

Finally, divide 4 into 33: it goes 8 times, witha remainder of 1.

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When dividing by a number bigger than 10, it is usually easier to set theworking out as a long division instead. The next example reminds you how thisis done.

EXAMPLE


SOLUTION

13��3�3�0�2�

213��3�3�0�2�

267

213��3�3�0�2�

2670

2513��3�3�0�2�

267065

5

25413��3�3�0�2�

267065

52520

So 3302 � 13 � 254 exactly


Bring down the next digit, 0 in this case, tomake the 7 up to 70.

13 divides into 70 five times, with remainder 5.

Finally, bring down the digit 2 to make 52.13 divides into 52 exactly 4 times, with noremainder.

Begin by setting up the problem using division bracket notation.

13 will not divide into 3, so divide 13 into 33.This goes 2 times, with remainder 7.

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1.4 Positive and negative integers

It is often convenient to visualise positive and negative whole numbers, orintegers, placed in order along a number line. The positive integers run to theright of zero, and negative integers to the left:

Mathematicians describe numbers on the right of the number line as beinglarger than the numbers on the left. This makes sense for positive numbers,where 6 is obviously bigger than 4, for example, but care must be taken withnegative numbers. 4 is bigger than �6, for example, and �8 is smaller than �7.

You need to be able to carry out basic arithmetic using positive and negativenumbers, with and without a calculator. Many calculators carry two types ofminus sign key: one for marking a number as negative, and another for theprocess of subtraction. So, in a calculation such as �6 � 5, you have to startwith the quantity �6 and then subtract 5. Subtraction means moving to the lefton the number line, so the answer is �6 � 5 � �11.


EXERCISE 1.3Use short division to work out the answers to these calculations. (Four of them should leave remainders.)

1 329 � 7 2 977 � 5 3 2686 � 9

4 28 845 � 3 5 1530 � 6 6 2328 � 8

7 1090 � 4 8 400 � 7

Use long division to work out the answers to these problems. Show your working clearly. (Only the last two should leave remainders.)

9 7684 � 17 10 7581 � 19 11 3315 � 15

12 4956 � 21 13 5771 � 29 14 3600 � 25

15 7890 � 23 16 3250 � 24

17 750 grams of chocolate is shared out equally between 6 people. How much does each one receive?

18 In a lottery draw the prize of £3250 is shared equally between 13 winners. How much does each receive?

19 Seven children share 100 sweets in as fair a way as possible. How many sweets does each child receive?

20 On a school trip there are 16 teachers and 180 children. The teachers divide the children up into equal-sized groups, as nearly as is possible, with one group per teacher. How many children are in each group?

Smaller at this end… … larger at this end.

�11 �10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7 8 9 10 11

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Take care when two minus signs are involved: the rule that ‘two minuses makea plus’ is not always trustworthy. For example, �3 � �5 � �8 (two minusesmake even more minus!), whereas �3 � �5 � �3 � 5 � 2. So two adjacentminus signs are equivalent to a single plus sign.

If two adjacent signs are the same: � � or � � then the overall sign is positive.

And if the signs are different: � � or � � then the overall sign is negative.

EXAMPLE

Without using a calculator, work out the values of:a) 6 � 9 b) �4 � 5c) �8 � �3 d) 5 � �6

SOLUTION

a) 6 � 9 � �3 b) �4 � 5 � 1

c) �8 � �3 � �11 d) 5 � �6 � 5 � 6 � 11

When multiplying or dividing with positive or negative numbers, it is usuallysimplest to ignore the minus signs while you work out the numerical value ofthe answer. Then restore the sign at the end.

If an odd number of negative numbers is multiplied or divided, the answer willbe negative.

If an even number of minus signs is involved, the answer will be positive.

EXAMPLE

Without using a calculator, work out the values of:a) (�5) � (4) b) (�4) � (�3)c) (�8) � (�2) d) 5 � (�4) � (�2)

SOLUTION

a) (�5) � (4) � �20b) (�4) � (�3) � 12c) (�8) � (�2) � 4d) 5 � (�4) � (�2) � 40


EXERCISE 1.4Without using a calculator, work out the answers to the following:

1 4 � (�6) 2 6 � (�3) 3 �3 � (�2)

4 2 � (�1) 5 �4 � 6 6 �4 � (�5)

7 �8 � 13 8 �3 � �15 9 (5) � �5

5 � 4 � 2 � 40 and there are two minussigns, so the answer is positive.

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1.5 Factors, multiples and primes

You will remember these definitions from earlier work:

A multiple of a number is the result of multiplying it by a whole number.

The multiples of 4 are 4, 8, 12, 16,…

A factor of a number is a whole number that divides exactly into it, with noremainder.

The factors of 12 are 1, 2, 3, 4, 6, 12.

A prime number is a whole number with exactly two factors, namely 1 anditself. The number 1 is not normally considered to be prime, so the primenumbers are 2, 3, 5, 7, 11,…

If a large number is not prime, it can be written as the product of a set of primefactors in a unique way. For example, 12 can be written as 2 � 2 � 3.

A factor tree is a good way of breaking a large number into its prime factors.The next example shows how this is done.

EXAMPLE

Write the number 180 as a product of its prime factors.

SOLUTION


180

18 10

Begin by splitting the 180 into a product of two parts. You coulduse 2 times 90, or 4 times 45, or 9 times 20, for example. Theresult at the end will be the same in any case. Here we begin byusing 18 times 10.

Since neither 18 nor 10 is a prime number, repeat the factorisingprocess.

10 5 � �5 11 6 � �2 12 �3 �4

13 4 � �8 14 �10 � �1 15 3 � �6

16 �4 � �5 17 �2 � 8 18 12 � �6

19 �18 � 3 20 �36 � �3

21 Arrange these in order of size, smallest first: 8, 3, �5, �1, 0.

22 Arrange these in order of size, largest first: 12, �13, 5, 9, �4.

23 What number lies midway between �4 and 12?

24 What number lies one-third of the way from �10 to 2?

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Thus 180 � 2 � 2 � 3 � 3 � 5� 22 � 32 � 5

1.6 Highest common factor, HCF

Consider the numbers 12 and 20. The number 2 is a factor of 12, and 2 is also afactor of 20. Thus 2 is said to be a common factor of 12 and 20.

Likewise, the number 4 is also a factor of both 12 and 20, so 4 is also acommon factor of 12 and 20.


EXERCISE 1.5

1 List all the prime numbers from 1 to 40 inclusive. You should find that there are 12 such prime numbers altogether.

2 Use your result from question 1 to help answer these questions:a) How many primes are there between 20 and 40 inclusive?b) What is the next prime number above 31?c) Find two prime numbers that multiply together to make 403.d) Write 91 as a product of two prime factors.

3 Use the factor tree method to obtain the prime factorisation of: a) 80 b) 90 c) 450

4 Use the factor tree method to obtain the prime factorisation of: a) 36 b) 81 c) 144What do you notice about all three of your answers?

5 When 56 is written as a product of primes, the result is 2a � b where a and b are positive integers. Find the values of a and b.

180

18 10

5229

180

18 10

52

33

29

18 has been broken down into 9 times 2, and 10 into 2 times 5.The 2s and the 5 are prime, so they are circled and the tree stopsthere.

The 9 is not prime, so the process can continue.

The factor tree stops growing when all the branchesend in circled prime numbers.

22 means the factor 2 is used twice (two squared). If it had beenused three times, you would write 23 (two cubed).

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It turns out that 12 and 20 have no common factor larger than this, so 4 is saidto be the highest common factor (HCF) of 12 and 20. You can check that 4 really is the highest common factor by writing 12 as 4 � 3 and 20 as 4 � 5;the 3 and 5 share no further factors.

EXAMPLE

Find the highest common factor (HCF) of 30 and 80.

SOLUTION

By inspection, it looks as if the highest common factor may well be 10.

Check: 30 � 10 � 3, and 80 � 10 � 8

and clearly 3 and 8 have no further factors in common.So HCF of 30 and 80 is 10

There is an alternative, more formal, method for finding highest commonfactors. It requires the use of prime factorisation.

EXAMPLE

Use prime factorisation to find the highest common factor of 30 and 80.

SOLUTION

By the factor tree method: 30 � 2 � 3 � 5

Similarly, 80 � 24 � 5

So HCF of 30 and 80 � 2 � 5� 10

The prime factorisation method involves a lot of steps, but it is particularlyeffective when working with larger numbers, as in this next example.

EXAMPLE

Use prime factorisation to find the highest common factor of 96 and 156.

SOLUTION

By the factor tree method: 96 � 25 � 3and 156 � 22 � 3 � 13

HCF of 96 and 156 � 22 � 3� 4 � 3� 12


By inspection means that you can just spot theanswer by eye, without any formal working.

Look at the 2’s: 30 has one of them, 80 has four. Pick the lower number: one 2

Look at the 3’s: 30 has one of them, but 80 has none. Pick the lower number: no 3s

Look at the 5’s: 30 has one of them, and 80 has one. Pick the lower number: one 5

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It is important to be able to use the prime factorisation method in case it appearsas an IGCSE examination question. You might like to try this ingeniousalternative approach. A Greek mathematician named Euclid used it 3500 yearsago, so it is often known as Euclid’s method.

EXAMPLE

Use Euclid’s method to find the HCF of 96 and 156.

SOLUTION

[96, 156] → [60, 96] → [36, 60] → [24, 36] → [12, 24] → [12, 12]

So HCF (96, 156) � 12

1.7 Lowest common multiple (LCM)

Consider the numbers 15 and 20.

The multiples of 15 are 15, 30, 45, 60, 75,…

The multiples of 20 are 20, 40, 60, 80,…

Any multiple that occurs in both lists is called a common multiple.

The smallest of these is the lowest common multiple (LCM). In this example,the LCM is 60.

There are several methods for finding lowest common multiples. As withhighest common factors, one of these methods is based on prime factorisation.


EXERCISE 1.6

1 Use the method of inspection to write down the highest common factor of each pair of numbers. Check your result in each case.a) 12 and 18 b) 45 and 60 c) 22 and 33d) 27 and 45 e) 8 and 27 f) 26 and 130

2 Write each of the following numbers as the product of prime factors. Hence find the highest commonfactor of each pair of numbers. a) 20 and 32 b) 36 and 60 c) 80 and 180d) 72 and 108 e) 120 and 195 f) 144 and 360

3 Use Euclid’s method to find the highest common factor of each pair of numbers. a) 12 and 30 b) 24 and 36 c) 96 and 120d) 90 and 140 e) 78 and 102 f) 48 and 70

Begin by writing the twonumbers in a square bracket.

Stop when both numbersare equal.

Each new bracket contains the smaller ofthe two numbers, and their difference.

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EXAMPLE

Find the lowest common multiple of 48 and 180.

SOLUTION

First, find the prime factors of each number using a factor tree if necessary.

48 � 24 � 3180 � 22 � 32 � 5

Look at the powers of 2:

48 � 24 � 3180 � 22 � 32 � 5

Next, the powers of 3:

48 � 24 � 3180 � 22 � 32 � 5

Finally, the powers of 5:

48 � 24 � 3180 � 22 � 32 � 5

Putting all of this together:

LCM of 48 and 180 � 24 � 32 � 5� 16 � 9 � 5� 144 � 5� 720

An alternative method is based on the fact that the product of the LCM and theHCF is the same as the product of the two original numbers. This gives thefollowing result:

LCM of a and b ��HCF

a

o

�

f a

b

and b�

This can be quite a quick method if the HCF is easy to spot.

EXAMPLE

Find the lowest common multiple of 70 and 110.

SOLUTION

By inspection, HCF is 10

So:

LCM � �70 �

10

110�

� 7 � 110� 770


There are 4 factors of 2 in 48, but only 2in 180. Pick the higher of these: 4

There is 1 factor of 3 in 48, but 2 in 180. Pick the higher of these: 2

There is no factor of 5 in 48, but 1 in 180. Pick the higher of these: 1

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EXERCISE 1.7

Find the lowest common multiple (LCM) of each of these pairs of numbers. You may use whichever methodyou prefer.

1 12 and 20 2 16 and 26 3 18 and 45

4 25 and 40 5 36 and 48 6 6 and 20

7 14 and 22 8 30 and 50 9 36 and 60

10 44 and 55 11 16 and 36 12 28 and 42

13 18 and 20 14 14 and 30 15 27 and 36

16 33 and 55

17 a) Write 60 and 84 as products of their prime factors.b) Hence find the LCM of 60 and 84.

18 a) Write 66 and 99 as products of their prime factors.b) Hence find the LCM of 66 and 99.c) Find also the HCF of 66 and 99.

19 a) Write 10, 36 and 56 as products of their prime factors.b) Work out the Highest Common Factor, HCF, of 10, 36 and 56. c) Work out the Lowest Common Multiple, LCM, of 10, 36 and 56.

20 a) Write 40, 48 and 600 as products of their prime factors.b) Work out the Highest Common Factor, HCF, of 40, 48 and 600.c) Work out the Lowest Common Multiple, LCM, of 40, 48 and 600.

It is also possible to find the HCF and LCM of three (or more) numbers. Theprime factorisation method remains valid here, but other shortcut methods canfail. This example shows you how to adapt the factorisation method when thereare three numbers.

EXAMPLE

Find the HCF and LCM of 16, 24 and 28.

SOLUTION

Write these as products of prime factors:

16 � 24

24 � 23 � 3

28 � 22 � 7

HCF of 16, 24 and 28 is 22 � 4

LCM of 16, 24 and 28 is 24 � 3 � 7 � 16 � 21 � 336

The lowest number of 2s from 24 or 23 or22 is 22

The highest number of 2s from 24 or23 or 22 is 24

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REVIEW EXERCISE 1Work out the answers to these arithmetic problems, using mental methods. Written working not allowed!

1 315 � 198 2 467 � 99 3 17 � 88 � 83

4 455 � 379 � 145 5 1005 � 997 6 43 � 11

7 599 � 3 8 396 � 4 9 456 � 12

10 53 � 7 � 53 � 3

Use pencil and paper methods (not a calculator) to work out the answers to these arithmetic problems.

11 866 � 372 12 946 � 268 13 124 � 7

14 144 � 23 15 44 � 77 16 651 � 37

17 2484 � 9 18 6812 � 13 19 7854 � 21

20 1000 � 16

Work out the answers to these problems using negative numbers. Do not use a calculator.

21 (�7) � (�14) 22 6 � (�3) 23 (�10) � (�13)

24 12 � �9 25 13 � �6 26 �5 � �8

27 �144 � 16 28 256 � (�8) 29 �7 � �4

30 (�3)3

31 Use a factor tree to find the prime factorisation of:a) 70 b) 124 c) 96 d) 240

32 a) Find the Highest Common Factor (HCF) of 24 and 56.b) Find the Lowest Common Multiple (LCM) of 24 and 56.

33 a) Write down the Highest Common Factor (HCF) of 20 and 22.b) Hence find the Lowest Common Multiple (LCM) of 20 and 22.

34 a) Write 360 in the form 2a � 3b � 5c

b) Write 24 � 32 � 5 as an ordinary number.

21 Virginia has two friends who regularly go round to her house to play. Joan goes round once every 4 days and India goes round once every 5 days. How often are both friends at Virginia’s house together?

22 Eddie owns three motorcycles. He cleans the Harley once every 8 days, the Honda once every 10 days andthe Kawasaki once every 15 days. Today he cleaned all three motorcycles. When will he next clean allthree motorcycles on the same day?

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KEY POINTS

1 Mental methods can be used for simple arithmetic problems. When adding upstrings of whole numbers, look for combinations that add up to multiples of 10.

2 Harder addition and subtraction problems require formal pencil and paper methods.Make sure that you know how to perform these accurately.

3 Simple multiplication problems may be done mentally or by short multiplication. Forharder problems, you need to be able to perform long multiplication reliably. If youfind the traditional columns method awkward, consider using the box methodinstead – both methods are acceptable to the IGCSE examiner.

4 Long division is probably the hardest arithmetic process you will need to master. The traditional columns method is probably the best method – there are alternatives,but they can be clumsy to use. If you have a long division by 23, say, then it may behelpful to write out the multiples 23, 46, 69, …, 230 before you start.

5 Exam questions may require you to manipulate and order negative numbers.Remember to treat the ‘two minuses make a plus’ rule with care, for example,�2 � �3 � 6, but �2 � �3 � �5.

6 Non-prime whole numbers may be written as a product of primes, using the factortree method. This leads to a powerful method of working out the Highest CommonFactor or Lowest Common Multiple of two numbers.

7 Sometimes you may be able to spot HCFs or LCMs by inspection. This result mighthelp you to check them:

LCM of a and b ��HCF

a

of�

a

b

and b�

35 Who is right? Explain carefully.

Chuck Lilian

36 Pens cost 25p each. Mr Smith spends £120 on pens. Work out the number of pens he gets for £120. [Edexcel]

37 The number 1104 can be written as 3 � 2c � d, where c is a whole number and d is a prime number. Work out the value of c and the value of d. [Edexcel]

38 a) Express 72 and 96 as products of their prime factors.b) Use your answer to part a) to work out the Highest Common Factor of 72 and 96. [Edexcel]

If the HCF of twonumbers is 1, then theymust both be primes.

Not necessarily true.

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Internet Challenge 1 �Prime timeHere are some questions about prime numbers. You may use the internet to help you research some of theanswers.

1 Find a list of all the prime numbers between 1 and 100, and print it out. How many prime numbers arethere between 1 and 100?

2 Find a list of all the prime numbers between 1 and 1000. How many prime numbers are there between 1 and 1000?

Compare your answers to questions 1 and 2. Does it appear that prime numbers occur less often as you goup to larger numbers?

3 Why is 1 not normally considered to be prime?

4 How many Prime Ministers has the UK had? Is this a prime number?

5 Is there an infinite number of prime numbers?

6 What is the largest known prime number?

7 Is there a formula for finding prime numbers?

8 Where is the Prime Meridian?

9 Find out how the Sieve of Eratosthenes works, and use it to make your own list of all the primes up to100. Check your list by comparing it with the list you found in question 1.

10 Some primes occur in adjacent pairs, which are consecutive odd integers, for example, 11 and 13, or 29 and 31. Find some higher examples of adjacent prime pairs. How many such pairs are there?

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