CHAPTER 1 Primes, Highest Common Factor and Lowest Common Multiple tive data that are transferred over the Internet, such as credit card numbers and passwords, have to be encrypted. In 1978, Ronald Rivest, Adi Shamir and Leonard Adleman publicly described the RSA algorithm, which isthe basis for public key cryptography. RSA provides a ‘method to ensure the secure encryption of data that even the most advanced ‘computers will take years to crack. It makes use of a complicated theorem involving a type of numbers called prime numbers. Prime numbers can be said to be the building blocks of all whole numbers ‘greater than 1. Every whole number {greater than 1 is either a prime or a unique product of primes! This has various applications in both mathematics and the real world. In this chapter, we are going to explore how whole numbers can be broken down into its building blocks. Learning Outcomes What will we learn in this chapter? + What prime and composite numbers are + How to find the square root ofa perfect square and the cube root of a perfect cube + Why highest common factor (HCE) and lowest common multiple (LCM) have useful applications in real life ae G Dipindai dengan CamScannerAlbert has a vanguard sheet with a length of 64 em and a breadth of 48 cm. He wants to cut it into squares that are, big as possible, without any leftover vanguard sheet. (@) Whatis the length of each square? (ii) How many squares can he cut altogether? In this chapter, we are going to learn about prime numbers, the highest common factor (HCF) and the lowest ‘common multiple (LCM), which can help us solve these kinds of problems, @ Prime numbers A. Whole numbers and factors (Recap) Let us recap on what we know about whole numbers and factors. What are the missing numbers below? Examples of whole numbers are 0,1, 2,1), 4,1, 6, ... Even numbers are whole numbers that are divisible by 2, ¢.g. 0, 2,4, (ND, 12,14, (Odd numbers are whole numbers that are not divisible by 2, -g. 13,5, Can you find the factors of 18? Ho... 3,0 Is =1x18 2~x® xe ‘Therefore, the factors of 18 are 1, 2, {Wi}, 6, and 18. Ein Is 18 divisible by each ofits factors? ‘Anumber nis divisible t another number p if ther no remainder when 7 is byp. B. Classifying whole numbers paaeeeaeee DA ‘Whole numbers can be divided (or classified) into two groups: even numbers and odd numbers. ‘Another way to classify whole numbers is to group them by the number of factors they have. 2 ‘CHAPTER ‘Primes Highest Common Factor and Lowest Com & Dipindai dengan CamScanneresa Classifying whole numbers 1. Find the factors of the numbers in Table 1.1. [Namba 7 Working” Factors] [Number Working Factors) 1 Lisdivisble by tonly 1 " 2 2=1x2 12 2 3 1B 4 24 4 5 15, 6 16 7 3 9 7 18 18=1%18=2%9=3%6 1,2,3,6,91 19 10 20 ‘Table 1d 2, Classify the numbers in Table 1.1 into 3 groups. Group A contains a number with exactly 1 factor: Group B contains numbers with exactly 2 different factors: Group C contains numbers with more than 2 different factors: 3, Is 0 divisible by 1, 2,3, 4, ete.? How many factors does 0 have? C. Prime numbers and composite numbers In the above Investigation, the number in Group A does not have a name. ‘The numbers in Group B are known as prime numbers (or primes). ‘The numbers in Group C are called composite numbers. Composite numbers are composed (or made up) of the product of at least two primes, e.g.6 = 2x3 and 18=2x3x3. 1, Explain why 0 and 1 are neither prime nor composite. 2. Yi Hao says that ifa whole number is not prime, then it must be composite. Do you agree? Explain your answer. Primes, Highest Common Factor and Lowest Common Multiple CHAPTE Dipindai dengan CamScannerSe ao In Table 1.1 in the Investigation on page 3, we identified prime numbers by finding all the factors of the num. ‘An easier way to sieve out prime numbers less than or equal to 100 is called the Sieve of Eratosthenes. Sieve of Eratosthenes 1. We will sieve out the prime numbers in Fig. 1.1 by circling those which are prime and crossing out those y are not. Follow the instructions below. (a) Cross out 1. (b) Circle 2. Cross out all the other multiples of 2 (because they are not primes: why?). (€) The next number that is not crossed out, i. 3, is a prime. Circle 3. Cross out all the other multiples of 3. (d) The next number that is not crossed out, i.e. 5, is a prime. Circle 5. ‘Cross out all the other multiples of 5. (e) Continue doing this until all the numbers have either been circled or crossed out. 1°, 93. Sh ets ery 8.9 = 10 aa 12 Bat Sloane? 18, 19 «= (20 2 220:« 23 A506 27 288 2980 BY, 20933) © S45 gS s eso a7? 38/39 40 41 42 43 44 45 46 «470 «48 4950 Sl), 52. 153) 154) 551756 miai57= -'58: -'59. 60 OU 62, 763. GA: 16S 66" 167" 08: 69 «70 7 472 873) 474,75) 176) 2771178) 79) 180) Bl. i82 83 86 185 86. .:87. -(88in, 89- 90 The ability to code or write, computer program is a useés “se oe 2 ee ee life skill. Although it is bere, Fig. 1.1 the scope of this textbook» teach you how to code, ifywe have some experience with 2. Answer the following questions. illea dots ceedings (a) What is the smallest prime number? in this textbook. Here's one (b) What is the largest prime number less than or equal to 100? NG arog ta pace (©) How many prime numbers are less than or equal to 100? antiie ey (d) Is every odd number a prime number? Explain. MO checwiee, if you sre tamer. (e) Is every even number a composite number? Explain. to learn, refer to Invitation () Fora prime number greater than 5, what can the last digit be? Explain. eee een Ucaeeennt & Dipindai dengan CamScannerProduct of prime numbers Can the product of two prime numbers be (a) an odd number? (b)_an even number? (©) prime number? Explain your reasoning or give a counterexample. D. Trial division ‘To test whether a number is prime or composite, we have to learn a new concept called square root. For example, 5 x 5 = 25. We say that the square of 5, or 5 squared, is 25. I'we do the reverse, we get V25 = 5, We say that the square root of 25 is 5. What is the value of Y9 and of Vi6 ? What about 47 ? Its value will not be a whole number. Why? Use a calculator to evaluate V/47 by pressing the [NEJM key. Did you get 6.9 (to 1 d.p.)? 1 6:9 (to 1 dp.) means the answer {69s correct to one decimal place. ‘To find out whether a number is prime or composite, we check if it is divisible by all the prime numbers less than or equal to its square root. Why? ‘This method is called trial division. Worked Example 1 shows how we can test whether a number is prime or composite. ‘Test for prime number pad Explain whether each of the following numbers is prime or composite. : (a) 387 (b) 997 *Solution (a) Since 387 is divisible by 3, then 387 isa composite number. (a) A numberis compsiteif isdvisible by any prime factor (The (b) /997 = 31.6 (to 1 d.p.), so the largest prime number less atic sca helpful to check whether a than or equal to 997 is 31. Ro mns ioe Since 997 is not divisible by any of the prime numbers Annumber is divisible by 3if 2,3,5, 7,11, 13, 17, 19, 23, 29 and 31, then 997 is a and only ithe sum ofthe (©) Anumber i prime ifitis ‘not divisible by all the prime number ss than or equal othe square rot of the number. Primes, Highest Common Factor and Lowest Common Multiple ‘CHAPTER 5 ‘ & Dipindai dengan CamScanner‘Are $3 and 1607 prime or composi 22 Inthis game, a policeman is chasing a thief. The policeman «can only step on tiles with prime numbers, Shade the correct tle to trace the path that he takes to catch the thief | ra fm fio) fa) so | } mmm — a) ou | 183 eo 7 o | | me | al} 7 | i | ibe ie n | afi] | 3 | - u wos | 9 | mn | 37 6 dots can be arranged in a rectangular array in two different ways (se Fig. 12). coo eccce cee Ly-6 2bys Fig. 12 (a) Twin primes are m= numbers that differ by such as 5 and 7. List typ ‘other pairs of tin ping (b) Cousin primes ate prin, numbers that differ by suchas and 11, List ty ‘other pairs of cousin prin, (©) Sexy primes are prime numbers that differ by such a8 2011 and 2017. two other paits of sexy primes 1 Arrange each of the following numbers of dots in a rectangular array in as many different ways as possible. How many different ways are there for each number of dots? (i) 4dots (ii) 8dots (ii) 12 dots iv) Sots. (W) 7 dots, 2. Other than by guess and check, is there a faster method to do Question 1? (vi) 1 dot 3. What do you notice about the numbers of dots that have only one arrangement? Why is this so? 4, What do you notice about the numbers of dots that have more than one arrangement? Why is this so? Problem involving prime number Rance) eae *Solution 2 prime number, then 1 and 13 ae its only aptgel+id=M4 Ifp and q are whole numbers such that p x q= 13, find the value of p+ g. Explain your answe = does not matter whether p (rq) is tor 13 because weos! ‘want to find the value of p +4 value ofp + q. Explain your answer 2. If nisa whole number such that n x (n+ 28) i prime number, find the prime number. Explain your answer. Band 31 are prime numbes with reversed digits. Name snother par of prime numbe: with reversed digits. ‘CHAPTER Primes Hight Common Factor nd Lowest Connon Mo & Dipindai dengan CamScannerE. Interesting facts and real-life applications of prime numbers Sees) Interesting facts about prime numbers 1. (i) How many prime numbers are there? ii). Search the Internet for ‘First 1 Million Primes’. What isthe 1 000 000" prime number? 2. (i) Since there are infinitely many primes, there is no largest prime number. However, the largest known prime number (at the time of printing) was found by Patrick Laroche on. 7 Dec 2018 and it contains 24 862 048 digits. [one newspaper page can contain 30 000 digits, how many newspaper pages are needed to print this prime number? (Gi) The size of the text fle containing this prime (pure text only) is 25 MB. Search “Largest Known Prime’ on the Internet, What is its last digit? ‘The largest known prime is the 51" known Mersenne prime and has a formula. Search the Internet for “Mersenne primes’ and learn about Mersenne numbers and primes. For many centuries, mathematicians studied prime numbers in a branch of mathematics called Number Theory out of interest. There were no real-life applications of prime numbers until the invention of computers when there was a need to encrypt sensitive data transmitted over the Internet, As explained in the Chapter Opener, RSA public-key cryptography makes use of a complicated theorem involving prime numbers to encode data securely. If the prime numbers chosen are large, it will take even the most sophisticated computers many years to crack the code. ‘What would have happened if mathematicians had not studied prime numbers out of interest? F. Index notation We have learnt that cm* is the unit for area and cm’ is the unit for volume. ‘The area of a square with length 5 cm is 5 cm x 5 cm = 25 em? 5 x5 can also be written as 5*, which is read as‘5 squared’. ‘The volume ofa cube with length 5 cm is 5 cm x 5 em x 5 em = 125 cm’. 5x5 x5 =5%, which is read as‘5 cubed’. What about 5 x 5x 5x 5? We can write 5 x 5 x 5 x 5 as 5*, which is read as'5 to the power of 4’, where 5 is called the base and 4 is called the index (plural: indices). Notations The index notation isa ‘mathematical notation | to represent the operat 5t is called the index notation of 5 x 5x 5x5. Write 3 x 3 x 3x 3x3 x 3x3 x3 in index notation: {| How is the index notation useful? “rltiplying by itsl? to ‘manner Ti For example, instead of writing that the speed of light is about 300 000 000 m/s, en pegeaee we can write this more concisely as 3 x 10" mis of numbers and multip signs, or so many zero: Primes, Highest Common Factor and Lowest Common Multiple CHAPTER & Dipindai dengan CamScannerCe G. Prime factorisation Consider composite number, eg, 18. =. expressed as a product of prime factors as shown: ‘The Fundamental Theoren, Itcanbe product of prime factors as shown aerate) 18 =2x3%3 ‘whole number Ptr thy x¥ Iseither a prime number canbe expressed aur “The process of expressing 18 asa product of ts prime factors is called the rt fs in, ol Iver there only ne ——— ihe oder ofthe prea Do not confuse the prime factorisation of 18 with finding the factors of 18 does not matter Inother words. rime nu, I= 1x 18=2x9=3%6, ate the bling blocks oy numbers greater than 1 ‘Notice that the factors of 18 are 1, 2, 3,6, 9 and 18, which are not necessari prime factors N Nan) Cee jon of 60, leaving your answer in index notation. 7 30. divide 60 by 2 to obtain sn, | NS and writ: AK, w= | x 37S + Then vide 30 by 2h, 15,and write | =202x | =2x3x5 + Then, divide 15 by 3tocb, Sand write: = 224345 ‘Method 2: + Finally, check that Divide 60 by the smallest prime factor and continue the process 2*2%3*5 sequltog. until we obtain 1, 2.60 =2x2x3x5 start with smallest —» 2 60. rimefactor. > —| ~ 39 =— divide 60 by STs 2toget 30 5 | 5 1 +— divide until we obtain 1 X2x3x5 Px 3x5 1, Find the prime factorisation of 126, leaving your answer in index notation. 2, Express 792.as a product ofits prime factors. 3. (i) Express 2021 asa product of its prime factors. (i) Given that a and b are whole numbers such that ais less than b and a x b = 2021 write down all the possible pairs (a,b). CHAPTER Prins, Highest Common Factor and Lowest Common eo & Dipindai dengan CamScannerProblem involving prime factorisation ‘Vani uses 231 one-centimette cubes to make a cuboid. Fach side of the cuboid is longer than ind the dimensions ofthe cuboid Tem “solution 281 = 37x H,where 3, Zand 1 are prime numbi {the length of any ide of Since each side of the cuboid is longer than 1 em, éuibo ean be tem, then there then the dimensions of the cuboid are 3 cm by 7m by Hem, more than one possi WMedx7 et a1ea77 Imran uses 195 one-centi 1s to make a cuboid. x7eS) Each side of the cuboid is longer than 1 cm. ie aa! Find the dimensions of the cuboid, Wicca tee oy . Joyce uses 324 one-centimetre cubes to makea cuboid. answer. ec ‘The perimeter of the top of the cuboid is 18 em. Each side of the cuboid is longer than 2 cm. Find the height of the cuboid. How are a number's prime factors different from its factors? 2. What are some methods to find the prime factorisation of a number? 3. What have I learnt in this section that I am still unclear off Exercise @ Determine wether each ofthe fllowingis@ prime @) (i) Express 2026 as a product ofits prime factors. | eng or a composite number. (i) Given that a and b are whole numbers such 026, write (a) 87 (b) 67 that ais ess than b and a x b = © 73 (@) 91 down all the possible pairs (a, b), @ Ifp and q are whole numbers such that p x q= 37, @ Cheryl uses 273 one-centimetre cubes to make find the value of p + q. Explain your answer. a cuboid. Each side of the cuboid is longer than 1m. Find the dimensions of the cuboid. If is a whole number such that n x (n + 42) isa prime number, find the prime number. Explain (@))_ Find all the prime numbers in each ofthe following your answer. decades. (a) 2011-2020 (b) 2021-2030 Find the prime factorisation of each of the following numbers, leaving your answer in index (@Q)_ Ifa and bare whole numbers such that a x b= 2027 notation. find the value of a + b. Explain your answer. (a) 72 (b) 756 © 187 @ 630 d the prime factorisation of each of the followin; numbers, leaving your answer in index notation, (a) 8624 (b) 6804 > (©) 26163 (@ 196 000 ‘Primes, Hight Common Factor and Lowest Common Mule MAPTERT . oh & Dipindai dengan CamScannerExercise GB) Given that x and y are whole numbers such that xis Devi uses 504 one-centimetre cubes to make less than y and x x y = 2022, write down all the a cuboid. The perimeter of the top of the cuboid i, possible pairs (x, 9). 20 em, Each side of the cuboid is longer than 2 cm, Find a possible height of the cuboid. Ali wants to use 210 one-centimetre cubes to make a cuboid such that each of its sides is longer than Lem, There are 6 possible cuboids that he can ake. Find the dimensions of any 3 of them. Square roots and cube roots A. Squares and square roots In Section 1.1D, we learnt that 5* = 5 x 5 = 25; we say that the square of 5, or 5 squared, is 25. The reverse is J25 = 5; we say that the square root of 25s 5. Copy and complete the following: + Since 0? = 0 x 0=0, then JO = VOx0 = + Since 1? = 1x 1 = {then Ji = Vixi =. + Since 2° = (i x GB = GB. hen i - «+ Since 3? = (iB x (i = GE. then JB = 5x5 - GD 0, 1,4 and 9 are squares of whole numbers, and they are called perfect squares (or square numbers). What are the next three consecutive perfect squares? | All perfect squares can be written as r?, where the square root nis a whole number. Worked Example 5 shows how we can find the square root ofa perfect square using prime factorisation. Finding square root using prime factorisation Notations “The superscript represents the square of a number while the symbol J~ represents the: a root of a number. | “The diagram below represents the inves lationship between gra5x5=25 squared oS square root VIB = 5x5 =5 ‘This diagram is useful for illustrating an inverse relationship. Bees Apis Find J324 using prime factorisation. Sj "Solution 2_| 324 Method 1: 2 162 324 =2x2x3x3x3x3 3{ 81 = 2x33) x (2x33) Fora number tobe a perfect Bear =Qx3x3? Sauare the index of each prime actor must be even. Why? $18 o VBR = 2x33 ad a a =18 1 a CHAPTER ‘Primes, Highest Common Factor and Lowest Common Muli <¢ 10 i & Dipindai dengan CamScannerMethod 2: B24 =2KIKINIKIKG =Px3 Vind = v2? 2x2 =18 ind V784 using prime factorisation, 2. Given that the prime factorisation of 7056 is 2" 3? x 7°, find ¥7056 without using a calculator. B. Cubes and cube roots 5? = 5x 5x5 = 125; we say that the cube of 5, or 5 cubed, is 125. If we do the reverse, we get Vi25 = 5; we say that the cube root of 125 is 5. What is the value of {27 and of Y64 ? Copy and complete the following: + Since "= 0x 0x0=0, then {0 = YOXOx0 =0. he Vi = MT =H. + Since 2 = (3 < (HB < (HN = (QB. then 26 = (< = weahd =2°x 3? aeformust be a multiple of sa perfect cube. 12 oar G Dipindai dengan CamScannerGEREN 1) veep ; perfect square. Goldhach’s Conjcture states 3. ‘on-zet0 whol ber. Given that 15 x 135 xk thit'every even number greater non-zero whole number. Given that 15 135 3k fav svey ven number 1 factors to explain why 15 x 135 is a a is a perfect cube, write down the smallest value of k. {he sam of two peimes. For ii) p and gare both prime numbers. Find the values of —_ leaving your answer correct to 4 decimal paces When the evaluation involves a fracti "Solution Method 1: Sequence of calculator keys: GoaaoB0ceouce0eh@onnacaa 8 +50, rant to key brackets, you must press Method 2: setre peice Why Sequence of calculator keys: eueeee0coEenEgmsnunCe 8+N50 Fes 702096 (104 dp) FREEBIE «1. sea calculator to evaluate each ofthe following leaving your answer correct to 4 decimal places where necessary. 2 . 3x20 (@) 208-7 — ) Shes 2. Thearea ofa square poster is 987 cm*. Find the perimeter of the poster, leaving your answer correct to I decimal place. 3, Nadia has 2020 one-centimetre cubes. She makes the largest cube possible using some of the 2020 cubes. How many cubes does she have left over? HAPTER ‘Primes Highest Common Fator and Lowest Common Null G Dipindai dengan CamScanner1, In Worked Example 5, what is something new that I have learnt about perfect squares? 2, In Worked Example 6, what is something new that I have learnt about perfect cubes? 3. What have I learnt in this section that I am still unclear off Exercise Find each of the following using prime factorisation. (a) i764 (o) 576 (©) 2916 (a) J3136 Given the prime factorisation of each of the following numbers, find its square root without using a calculator. (@) 9801 =3'x1P () 35721 =3°x7? (c) 24336 = 2! x 3? 13? (a) 518.400 = 28x 3'x5* Find each of the following using prime factorisation. (@) 43375 (b) i728 (©) ¥5832 (@) {8000 @ Given the prime factrisation ofeach ofthe following numbers, find its cube root without using a calculator. (a) 21952=2%x7? (b) 46.656 = 26x 3° (0) 287496= 2x3 x1? (4) 1728000=2x 3x5? @ without using a calculator, estimate the value of cach of the following. @) J6 () V0 () ya8 (a) 730 Usea calculator to evaluate each of the following, leaving your answer correct to 4 decimal places where necessary. (a) 7-v36r+21 ——(b) ee (© Ve +faa3 Find the smallest non-zero whole number which can be multiplied by 112 to give (i asquare number, (ii) a cube number. A textbook is opened at random. Without using a calculator, find the pages the textbook is opened to, given that the product of the facing numbers is 420. Hint: 400 isa perfect square. The area ofa square photo frame is 250 cm’. Find the perimeter of the photo frame, leaving your answer correct to 1 decimal place. ‘The volume of a box in the shape of a cube is 2197 cm’, Find the area of one side of the box. GD Raju has 2020 one-centimetre square tiles. @w He makes the largest square possible using some ‘of the 2020 square tiles. How many square tiles does he have let over? Use prime factors to explain why 6 x 54 is a perfect square, kis non-zero whole number. Given that 6 x54 x kisa perfect cube, write down the smallest value of k. (iil) p and q are both prime numbers. Find the (i) values ofp and q so that oxsixt isaper cube Primes, Highest Common Factorand Ton CHAPTERT G Dipindai dengan CamScannerExercise | © wo expresss704 a the product ofits prime factors. (Ii) Find the smallest whole number ft such that (i) Given that 4224 9, where m and m are a isa cube number. ‘whole numbers and 1 is a large as possible, find the value of mand of n. Highest common factor and lowest common multiple A. Highest common factor (HCF) In primary school, we have learnt about factors and common factors, €g. 18=1x18 30=1%30 x9 2x15 x6, =3x10 Factors of 18: Factors of 30 5 x6 9, 8 10, 15, 30 + the common factors of 18 and 30 are 1,2,3and 6, fall the common factors of 18 and 30, the highest is 6. ‘We say thatthe highest common factor (HCF) of 18 and 30s 6. The lowest common factor of 18 ‘This method of finding the HF of two or more non-zero whole numbers is called the snd 20/ bln fac are listing method. tore noa-zero whole numbers What is the HCF of 504 and 588? neta 504 has 24 factors while 588 has 18 factors. Altogether, 504 and 588 have 12.common factors. “The listing method to find the HCF of $04 and 588 is tedious because it involves many factors and common factors. ‘We will now learn more efficient methods to find the HF of two or more numbers, Finding HCF of two numbers 23) Find the highest common factor of 18 and 30. Example crop) ‘soution Method I: Prime factorisation common primefactors common prime factor or 18-f2hy 30-{2lef3 fx 5 common factor is 3, ie. choose 3 with the HCE of 18 and 30=2 x3 2x34+—smaller index: =6 ‘ 16 ren Primes Highest Common Factor and Lowest Common Stull! & Dipindai dengan CamScannerMethod 2: Ladder method common [2] | 18,30 + divide 18 and 30 by 2to get 9 and 15 Pele factors 9,15 + divide 9 and 1S by 3 to get Sand 5 3 8 + stopdividin Mcreriseadsoet ts no common prime factors =6 1, Find the highest common factor of 56 and 84 using both methods. eres 2, Using the prime factorisation method, find the largest 2. Theat wholemunber eae whole number that isa factor of bot 5 pie hee eee Seer le number that is a factor of both 112 and 140. pete eee ee ea 3. The numbers 504 and 588, written as the products of their and 140. prime factors, are 504 = 2" x 3? x 7 and 588 = 2? x 3x7", Hence, explain why 84 is the greatest whole number that will divide both 504 and 588 exactly. \ding HCF of three numbers Find the HCF of 40, 60 and 100. arc) eu Ship “Solution ‘Method 1: Prime factorisation common factors are 2 and 5, ie, choose each of the common common prime factors prime factors with the smallest index: t HCE of 40, 60 and 100 = 2.x 2 x5 =20 ‘Method 2: Ladder method 2] | 40, 60, 100 <—divide 40, 60 and 100 by 2 to get 20, common — 30nd 50 prime <*|2| | 20, 30, 50 <—divide 20, 30 and 50 by 2 to get 10, 1: factors and 25 5] | 10, 15, 25 + divide 10, 15 and 25 by 5 to get 2,3 a : ands 2, 3, 5 ¢—stop dividing when there are no common prime factors ies, Highest Common Factor and Lowest Common Multiple current 7k & Dipindai dengan CamScannerid the HCF of 90, 135 and 2: ? Factors ‘Determine whether each of the following statements is true or false. Ifit is true, explain your reasoning. If itis False, give a counterexample. (a) 1€6 sa factor ofa number, then 2 and 3 are also factors ofthat number. () If and 3 are factors ofa number, then 2 x 3 = 6 is also a facto ofthat number. (6) 1f2 and 4 are factors ofa number, then 2 x 4 = 8 is also a factor ofthat number. (@) Iffisa factor of n, then ; isalso a factor of n. (6) If his the HCF ofp and q, then both p and g are divisible by h. B, Lowest common multiple (LCM) eal died aly In primary school, we have learnt about mi Multiples of 4: 4, 8, 16, 20, 28, 32, 40, Moultiplesof6: 6, 18, 30, 2 ©: the common multiples of 4 and 6 are 12, 24, 36, ... How many common multiples of 4 and 6 are there? p= What is the highest common multiple of 4 and 6% Thehighe common multiple of any two of more non-zeFO (fall the common multiples of 4 and 6, the lowest is 12 Solum avays ‘We say that the lowest common multiple (LCM) of 4 and 6 is 12. ‘This method of finding the LCM of two or more non-zero whole numbers i called the listing method, les and common multiples, eg. What is the LCM of 504 and 540? ‘The LCM is the 15% multiple of 504 and the 14% multiple of 40. It would be very tedious to list the first 15 multiples of 504 and the first 14 multiples of 540 to find the LCM. ‘We will now learn more efficient methods to find the LCM of two or more numbers. Sg warren Primes, Highest Common Factor and Lowest Common Mule r G Dipindai dengan CamScannerae Finding LCM ‘The product of 4 and 6, the LCM of 4 and 6 is 12. ‘We see that the LCM of 4 and 6 cannot be found by taking the product of all the prime factors in the prime factorisation of 4 and 6, ie. LEM # (2 2) x (2% 3), So why is the LOM equal to 12 (= 2x 2x3)? 24, is a common multiple of 4 and 6. However, using the listing method, we found that Since the LCM of 4 and 6 isa multiple of 4 and a multiple of 6, we get: Axh=6xk i axa Both sides of the equation contain a prime factorial highlighted in green) But the left-hand side (LHS) of the equation contains another prime factor’ (highlighted in blue), which the right-hand side (RHS) of the equation does not have, So k =/2\(since we want k to be the smallest). Now, the RHS of the equation contains a prime factorBl(highlighted in purple), which the LHS does not have. So h =B(since we want h to be the smallest). ¢ LOM of and 6 =B)x Bx From the above explanation, we derive the prime factori: shown below: jon method of finding the LCM of 4 and 6 as common prime factor: choose the common factor (i.e. 2) take only one with the higher index (why?) 6 6a2k3 v + LCM of 4 and 6 2x3 ‘The ladder method for finding the LCM of 4 and 6 is as follows: common prime —>| 4, 6 <—divide 4 and 6 by 2 to get 2 and3 factor |<— stop dividing when there are no common prime factors LCM of 4 and 6 =2x2xB) “ = 12 remaining factors Primes, Highest Common Factor and Lowest Common Multiple current 4 & Dipindai dengan CamScanner| eet] en ind the lowest common n “Solution Method 1: Prime factorisation common prime factors Method 2: Ladder method commo prime factors 30, 15, sinding LEM of two numbers vate ultiple of 30 and 36. choose each of the common printe Melo, i.e. 2 andl 3) with the higher index snd the remain 5) factor (ie 36-4 divide 30 and 36 by 210 get 15 and 18 18-4 divide 15 and 18 by 3 to get 5 and 6 _—6]=— stop dividing when there are LCM of 30 and 36 =2x3x5x6 eng 180 retaining factors no common prime factors 1, Find the lowest common multiple of 24 and 90 using both methods. number that is di Finding LCM of three numbers Find the LCM of 12, 18 and 56. “Solution Method 1: Prime factorisation common prime factors reo4 LCM of 12, 18 and 56 or 2, The numbers 120 and 126, written as the products of their prime factors, are 120 = 2° x 3 x 5 and 126 = 2.x 3 x 7. Hence, find the smallest non-zero whole ible by both 120 and 126. choose each of the common prime factors (ie. 2 and 3) with the highest index and the remaining factor (ie. 7) Dias x7 20 more CHAPTER Primes, Hi rst Common Factor and Lowest Common Multi & Dipindai dengan CamScannerMethod 2: Ladder method these 3 numbers have no start with the smallest i | 1, 18, 56 4| 3isa.common prime — factor of 3 and 9, so we divide 3 and 9 by 3 any. LCM of 12, 18 and 56 common prime factors, but 6 and 28 have a commmion prime factor 2,0 we divide 6 and 28 by 2 stop dividing when there are no ‘common prime factors between fo numbers Find the LOM of9, 30nd 108 Multiples Deter give a counterexample. (a) If6is a multiple of a number, then 12 is also a multiple of that number. (b) 1f12isa multiple of a number, then 6 is also a multiple of that number. (c) If18 isa multiple of a number, then 18 is divisible by that number. (@) If mis a multiple of n, then ™ is a whole number. n (e) Iflis the LCM of p and q, then ible by both p and q. Finding factors of number ine whether each of the following statements is true or false. Ifit is true, explain your reasoning. If it is false, fm List the factors of the LCM of ad ‘A number has exactly 8 factors, two of which are 6 and 27. List all the factors of the number. Suis 14 Be 6=2%3 27 e LCM of 6 and 27 = 2x 3° 6 and 27 to see if there are 8 factors. Since there ae 8 factors, then the number is the LCM of Gand 27. 54 21x54 Ifthereare less than 8 factors, cone ‘ry to multiply the LOM by = prime number to see if the =3x18 result has exactly 8 factors (see =6x9 Practise Now 14 Question 2).. Sometimes, itis possible + the number is 54 and its factors are 1,2, 3, 6,9, 18,27 and 54. _Ghtan more than one answer (see Exercise 1C Question 24). ‘Prinses, Highest Common Factor and Lowest Common Multiple CHAPTERT 21 & Dipindai dengan CamScanner=—_ °° ° °°» © Ea “A number has exactly 8 factors, two of which are 27 and 45. List all the factors of Poon Marc Can eee Rance) eee) 16 the number. 2. Anumber has exactly 8 factors, two of which are 4 and 20. List all the factors of the number. Finding number given another number and their LCM Find the smallest value of n such that the LM of n and 6 is 24. *Solution common prime factor: "This isa reverse question, vu take only one the prime factrisation metho backwards, “Thee isat most one common prime factor between m and, ‘The frat factor 2 for wil be 7 common with the fctor 2 fre, Wedo not use factor 3 since ve LCM =24=2%2x2x3 LOM=24=2x2x2x3 — wantntobe the smallest. The other two factors (2 and 2) sm=2X2x2=8 ust be the factors of 1, Find the smallest value of such that the LCM of n and 15 is 45. 2, The LCM of 9, 12 and kis 252. If kis odd, find all the possible values of k. Finding numbers given their HCF and LCM. ‘The highest common factor of two numbers is 56 ‘The lowest common multiple ofthese two numbers is 2520, Both numbers are greater than 56, Find the two numbers. *Solution We will use Pélya’s Problem Solving Model to guide us in solving this problem. Stage 1: Understand the problem This is a reverse question because the two numbers are not given but their HCF and LCM are given. Both numbers are also given to be greater than their HCF, so neither of them can be equal to the HCF (= 56). i22 ‘CHAPTER Primes, Highest Common Factor and Lowest Common Mule & Dipindai dengan CamScannerStage 2: Think of a plan ‘Since the HCP is a factor of each of the two numbers, then both numbers must contain the HF as a factor as shown: Let the two numbers be a and b. Since the LCM contains three more factors (ic. 8 and), we have to distribute these factors into a and b stich that: + the HCF is still587 ({.e. we cannot ‘give’ one@ to a and the other ¥ to b, or else the HCF will become 27); + both a and b are greater than the HCF (i.e. we cannot ‘give’ all the remaining factors to only @ orb; since we ‘give’BEto one of them, we have to give ito the other number), Stage 3: Carry out the plan LOM = 2520 = 2° x 38x 5x7 = (2x7) x Fx5 Let the two numbers be a and b. It does not matter whether we ‘give'3* toa ortob. the two numbers are 2? x 7x 3 = 504 and 2x7 5= Stage 4: Look back How can we check that the answer is correct? Method 1: Find the HCF and LCM of 280 and 504. Method 2: Use the fact that the product of the two numbers, ir HCI LCM, ie. ax b=HCF x LCM (why?) not be equal to HCE x LCM. Check: 280 x 504 = 141 120 See Challenge Yourself HCP x LCM = 56 x 2520 = 141 120 Question 5 ‘The lowest common multiple of these two numbers is 4410. poe Both numbers are greater than their highest common factor. ere Find the two numbers. 2. The numbers 240 and 252, written as the products of their prime factors, are 240 = 24 x 3 x Sand 252 = 2 x 3*x 7, Find (i) the smallest non-zero whole number 1 for which 240n is a multiple of 252, (ii) the smallest non-zero whole number m for which “ js factor of 252. Primes Highest Common Factor and Lowest Common Multiple urn 93 & Dipindai dengan CamScanner= 7 C. Real-life applications of HCF and LCM —— eee ‘We have learnt how prime numbers can help us find the HCF and LCM of two or more numbers. In this S€Ctigg ‘ill solve some real-life problems involving H1CF and LCM. Real-life problem involving LCM The lights on thre lightships ash at regular intervals. The frst ight ashes every 18 secy, the second every 30 seconds and the third every 40 seconds. The three lights flash togeth, GYyfp 1000p. Atwhat tine wi they nex ash ogee? *Solution EE, 18= 2x3 The LCM of the three tims, Nene pe =2x3x5, theimterval at which the li, 40= 2x5 Aas together + LOM of 18,30 and 40 = 2x 38x 5 = 360 360 seconds = 6 minutes <+the thee lights wll next fash together at 1.6 pam ‘Three bells toll at regular intervals of 15 minutes, 16 minutes and 36 minutes respectively. If they tll together at 2.00 pam, what time will they next toll together? 2. LiTing has thee pieces of rope measuring 140 cm, 168 cm and 210 cm. She wishes tocutthe three pieces of rope equally into smaller pieces without any leftover rope. ()_Whatis the greatest possible length of each of the smaller pieces of rope? (Gi) How many smaller pieces of rope can she cut altogether? Introduct ry Problem Set) “Now that you have learnt about HF and LCM, solve the Introductory Problem and discuss your solution with. your classmates. 1, What do I already know about common factors and common multiples that could help me understand what HCF and LCM are? +2. When faced with a real-life problem involving HF or LCM, how do tell whether to use HF or LCM to find the solution? & Dipindai dengan CamScanner\ 5 \ Exercise @ Find the highest common factor of each of the following sets of numbers. (a) 12and30 (b) 1Band91 (©) 126and 240 (a) 180and 450 () Mand3i (© Gtandat @© Aina the largest whole number that sa factor of both 156 and 168. @ Find the highest common factor of each of the following sets of numbers. (@) 15,60 and 75 (©) 336,392 and 504 (© 7,17 and23 (b) 360, 540, 882 (@)_ 77,91 and 143 (0 252, 588 and 840 Explain why 48 is the largest whole number that is a factor of 288, 336 and 480. Find the lowest common multiple of each of the following sets of numbers. (@) 45and60 (b) 42and462 (©) S4and 240 (d@) Land 19 (©) 27and32 () 78 and 352 Explain why 1040 is the smallest non-zero whole ‘number that is divisible by both 80 and 104. Find the lowest common multiple of each of the following sets of numbers. (@) 12,18 and81 (© 3,7and 11 (@) 26,52and 104 (b) 63,80 and 102 (d)_ 25,27 and 32 (91,143 and 169 Find the smallest non-zero whole number that is divisible by 315, 378 and 392. ‘Annumber has exactly 8 factors, two of which are 10 and 125. List all the factors of the number. A number has exactly 12 factors, two of wl 40 and 100, List all the factors of the number. ind the smallest value of k such that the LCM of kand 6 is 60. Primes, Highest Common Factor and Lowest Convaon Malte GB) The highest common factor of two numbers is The numbers 792 and 990, written as the products of their prime factors, are 792 = 2? 3? 11 an 990 = 2% 3? x5 x 11. Hence, explain why 198 is the greatest whole number that will divide both 792 and 990 exactly, @ the numbers 1080, 1188 and 1815, written as the products oftheir prime factors, are 1080 = 2° x 345, 1188 = 2? x 3° LI and 1815 = 3 x5 x I, Hence, find the greatest ‘whole number that will divide 1080, 1188 and 1815 exactly. ‘The numbers 176 and 342, written as the products ‘of their prime factors, are 176 = 2! 11 and 342 = 2.x 3#x 19. Hence, find the smallest non-zero whole number that is divisible by both 176 and 342, (BB) The numbers 120, 1276.nd 1450, written as the products oftheir prime factors, are | 5% 7, 1276=2?x 11 x29and x 5? x 29. Hence, find the smallest isible by 1120, 1450 = non-zero whole number that is 1276 and 1450. A number has exactly 8 factors, two of which are 4 and 26. List all the factors of the number. ‘The LCM of n, 28 and 49 is 588. If n is odd, find alll the possible values of n ‘The lowest common multiple of these two numbers is 12.600. Both numbers are greater than their highest common factor, Find the two numbers. B) (Express 1050. the product of its prime factors. (ii) ind to numbers, both greater than 40, that havea highest common factor of 21 and a lowest common multiple of 1050. current 95 & & Dipindai dengan CamScannerExercise (iia (8) shufen needs to pack 171 pens, 63 pencils and 27 erasers oqualy into identical git bags. Find (i) _ the largest number of gift bags that can be packed, (i) the number of each item in a git bag GAY sweets are sold in packs of 120 while mini chocolate bars are sold in packs of 18. Siti bought the same number of sweets as mini chocolate bars. Find the least number of packs of sweets that she could have bought. @ ‘Two race cars, Car X and Car Y, are at the starting point ofa 2-km track atthe same time, Car X and Car Y¥ make one lap every 60 s and every 80's respectively. @ How long, in seconds, wil it take for both cars tobe back atthe starting point atthe same time? (Gi) How long, in minutes, wil it take forthe faster car tobe 5 laps ahead of the slower car? (BB) Kumar wishes to cut the biggest posible squares from a rectangular sheet of paper without any leftover paper. The sheet of paper has. length of 65cm and a breadth of 50 cm. (i) Whatis the length of each square? (ii) How many squares can he cut altogether? Q the numbers 528 and 540, written as the products @ ‘he numbers 630 and 1248, written as the produ Find two numbers that each have exactly 16 factors, two of which are 8 and 12, ‘The LCM of m, 49 and 63 is 882. Find 6 possible values of m. of their prime factors, are 528 = 2'x 3x 1 and 540 = 2° 3? x 5, Hence, find (i) the smallest non-zero whole number h for which 528h isa multiple of 540, the smallest whole number k for which 228 a factor of 540. of their prime factors, are 630 = 2x 3?x 5 x 7 an 1248 = 283 x 13. Find i) the smallest non-zero whole number n for which 630n is a multiple of 1248, (i) the smallest whole number m for which is a factor of 630. ‘A class has between 30 to 40 students. Each boy i the class brings 15 chocolate bars for a class party, The chocolate bars are shared equally among the 20 girls of the class and their form teacher. There. are no leftovers. (i) How many students are there in the class? (ii) How many chocolate bars does their form. teacher receive? & Dipindai dengan CamScannerane [Eres JDOo In this chapter, we learnt about prime numbers, the building blocks of whole numbers greater than 1, We also learnt that whole numbers greater than 1 can be classified into two categories: prime numbers and composite numbers. Here are a few important lessons you must remember as we move forward: 1, Definitions in mathematics are important because they allow us to specify the properties of mathematical “objects, such as prime numbers. We must learn to understand and apply them as we examine new mathematical objects. Index notations help us write the prime factorisation of a number more concisely. Notations are important because they help us represent mathematical objects and their operations or relationships in a more concise and precise manner. We can use the prime factorisation of two or more numbers to find their highest common factor (HCF) and lowest common multiple (LCM). There are many useful applications of primes, HCF and LCM, both in mathematics and in the real world, ANIA AIA AAA AAA i 1. A prime number isa whole number that has exactly 2 different factors, | and itself. Examples of prime numbers are 2 and 7. + Give two other examples. (a) A composite number is a whole number that has more than 2 different factors. Examples of composite numbers are 6 and 15. + Give two other examples. (b) The process of expressing a composite number as a product of its prime factors is known as prime factorisation, e.g. 18 = 2x 3x3 (or2x3?). + Give another example of prime factorisation. 3. Whole numbers can be classified into three groups as shown below. Fill in the blanks. Whole numbers (0, 1,2, 3, | Neither prime nor Prime numbers Composite numbers composite has exactly 2 different factors has more than 2 different factors (Sand) (2,3,5,7, 1s.) (4, 6, 8,9, 10, ...). 4. A perfect square (or square number) is a whole number whose square root is also a whole number. An example of a perfect square is 25 because 25 = 5’, or 25 = 5. + Give two other examples of perfect squares and find their square roots. & Dipindai dengan CamScanners. A perfect cube ( ‘number) is a whole number whose cube root js also a whole number. An example ofa perfect cubes 125 because 125 =8'or V5 = wo other examples of perfect cubes ind find their cube roots ‘The highest common factor (HCE) of two or more numbers isthe largest factor that is common to all the numbers eg the HCE of 18 and 30 86 + Give another example the numbers, e.g. the LCM of 12, 18 and 56 is 504, + Giveanother example, ‘Think ofa real-life problem that you can use the HCE or the LCM to help you solve. 1, Explain whether each of the following numbers is 2 3. 5. prime or composite. (a) 753 (b) 757 Find each ofthe following using prime factorisation (a) is () ive Estimate the value of each ofthe following. @) VB (e) Yas ‘The numbers 840 and 8316, written as the products of their prime factors, are 840 = 2!x 3 x5 x7 and 8316 = 2 3x7 x LI. Hence, find (i) the greatest whole number that will divide both 840 and 8316 exactly, (i) the smallest non-zero whole number that is divisible by both 840 and 8316. ‘The lowest common multiple (LCM) of two or more numbers is the smallest multiple that is common to ali . Nadia needs to arrange 108 stalks of roses, 81 stalks of lilies and 54 stalks of orchids into identical baskets so that each type of flower is equally distributed among the baskets. Find (0) thelargest number of baskets that can be arranged, (ii) the number of each type of flower in a basket. ‘AtSA5 pam, Joyce, Bernard and Weiming are at the starting point ofa I-km circular path. Joyce takes. 15 minutes to walk 1 round, Bernard needs 360 seconds to run 1 round and Weiming cycles 2 rounds in 6 minutes. Find the time when all three of hem will next meet atthe starting point. ‘Vani and Li Ting work in different companies. ‘Vani has a day off every fourth day while Li Ting has a day off every sixth day, Vani's last day off was on 29 April while Li Ting's was on 1 May. (Note: Thereare 30 days in Apriland 31 days in May.) (i) When will they next have the same day off? ‘The LCM of 6, 12 and n is 660, Find all the possible values of n (ii) Subsequently, how often will they have the same day off s 28 ore Primes Highest Conspen Factor and Lowest Common Maiti & Dipindai dengan CamScannerET 9. The numbers 504 and 810, written as the products (i) Use prime factors to explain why 15 x 375 isa of their prime factors, are 504 = 2° x 3° 7 and perfect square. ESS | (ii) kisa non-zero whole number. Given that the smallest non-zero whole number n for 15 375 x kisa perfect cube, write down the which 504m isa multiple of 810, smallest value of k (ii) the smallest whole number m for which 224 (iii) pand q are both prime numbers. Find the m isa factor of 810. values of p and q so that isxarset isa perfect cube. Hints for Challenge Yourself are provided on page 205. 1. Anew school has 1000 students, and 1000 closed lockers numbered 1 to 1000. The first student opens all the 1000 lockers. The second student closes all lockers with numbers that are multiples of 2. . ‘The third student ‘reverses’ all lockers with numbers that are multiples of 3 — he closes the locker if it is open, and opens the locker ifit is closed. The fourth student reverses all lockers with numbers that are multiples of 4, and so forth until all the 1000 students reverse the relevant lockers, Which lockers will be left open in the end? Guiding questions (based on Pélya’s Problem Solving Mod Stage 1: Understand the problem (a) What do you understand by the term ‘reverses’ the locker? Stage 2: Think of a plan (b) Can you simplify the problem for, say, 10 lockers? Stage 3: Carry out the plan (©) What do you notice about the locker numbers that are left open in the end? Is there a pattern? (4) How do you know that the pattern will be true for 1000 lockers? Stage 4: Look back (e) How can you check if your answer is correct? 2. The figure shows a shape made up of three identical squares. Divide it into four identical parts, & Dipindai dengan CamScannerire shows the face h the numbers 1 to 1 (Which are the six adjacent numbers such that the sum of every pair of adjacent numbers for these six numbers isa prime number? clock (i) Rearrange the other six numbers so that the sum of every pair of adjacent numbers isa prime number. How many ways are there to do this “The diagonal of a 6-by-4 rectangle passes through 8 squares as shown in the figure. nd a formula for the number of squares passed through by a diagonal of a meby-n rectangle. (@ Find the HCF and LCM of 120 and 126. Gi)_ Show that the product of the HCF and LCM of 120 and 126 is equal to the product of 120 and 126. By looking at their prime factorisation, explain why this is so. (iii) Can you generalise the result in part (ii) for any two numbers? Explain your answer. (iv) Can you generalise the result in part for any three numbers? Explain your answer. (i) Find the least number of cuts required to cut 12 identical sausages so that they can be shared equally among 18 people. (Note: Each cut made to a sausage is considered one cut.) (i) Find the least number of cuts in terms of m and n, required to cut m identical sausages so that they can be shared equally among n people. ‘Yi Hao thinks of seven different non-zero whole numbers that are relatively prime to one another, i.e. the highest common factor of any two of them is 1. What is the least possible sum of these seven numbers? Find the largest number of composite numbers less than 2020 that are relatively prime to one another. AA fast food restaurant sells nuggets in boxes of three different types: small, medium and large. A small box contains 6 nuggets, a medium box 9 nuggets and a large box 20 nuggets. (@) Albert wants to buy exactly 41 nuggets. How many boxes of each type should he buy? (ii) Cheryl wants to buy exactly 22 nuggets. Is it possible to do so? Explain. (iii) What is the least number of nuggets, n, such that itis possible to buy exactly n nuggets, exactly m+ 1 nuggets, exactly +2 nuggets, and so forth? Explain. 30 0" Primes, Highest Common Factor and Lawest Goauaon Mite & Dipindai dengan CamScanner