STRAND B: NUMBER THEORY
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1 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors, HCF nd LCM B. Further Inde Nottion
2 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B Indices nd Fctors B. Squres, Cubes, Squre Roots nd Cube Roots When number is multiplied by itself, we sy tht the number hs been squred. For emple, squred mens 9. This is written s 9. We could lso sy tht 9 is the squre of. When number is cubed it is written down times nd multiplied. For emple cubed mens 8. This is written s 8. We could lso sy tht 8 is the cube of. Sometimes the reverse process is needed to nswer questions such s: Wht number squred gives The nswer would be. We sy tht is the squre root of, or write. Another question might be: Wht number cubed gives 8 The nswer would be. We would sy tht the cube root of 8 is. We could lso write 8. Worked Emple Find () 8 (b) (c). Use your nswers to find (d) (e) (f) () (b) (c) (d) 8 becuse 8 (e) becuse (f) becuse
3 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet Eercises. Find () (b) (c) (d) Use your nswers to find (e) (f) (g) 9 (h). Find () (b) (c) (d) 0 Use your nswers to find (e) (f) 000 (g) (h). Find () 0 (b) (c) (d) (e) 8 (f) 9 (g) (h) (i) 8 (j) 0 (k) 0 (l). Find () 00 (b) (c) 8 (d) (e) (f) 9. Use clcultor to find () (b) (c) (d) (e) (f) (g) 0 (h) Without clcultor, find (i) (j) 00 (k) 9 (l) (m) (n) 9 (o) (p). Find () + (b) (c) 0 + (d) + (e) (f) + (g) + 0 (h) + 8 Informtion On verge, humn hert bets times minute, 00 times n hour, times dy, times yer nd times for someone who lives 80 yers.
4 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B. Inde Nottion Inde nottion is very useful wy of writing epressions like in shorter formt. The bove could be written with inde nottion s. The smll number,, is clled the inde or power. Worked Emple Find () (b) (c) () (b) (c) 8 0 Worked Emple Find the missing number. () (b) (c) ( ) ( ) () 0 ( ) ( ) (b) (c) Note m n m+ n nd n m nm These rules pply whenever inde nottion is used.
5 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet Using these rules, 0 or So 0 Worked Emple Find () ( ) (b) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) (b) Note Eercises ( ) m n m n. Write ech of the following using inde nottion. () (b) (c) (d) (e) (f) 9 9 (g) (h) (i) (j) Find the vlue of ech of the following. () (b) (c) (d) 0 (e) 0 (f) (g) (h) (i) 8 (j) (k) 0 (l)
6 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet. Fill in the missing numbers. () (b) (c) (d) (e) (f) 9 (g) (h) (i) 0 (j) (k) (m) (l) (n) 0 (o) 8 (p) (q) (r) 0 (s) (t) 8 0 (u). Fill in the missing numbers. () (b) 8 (c) (d) (e) (f) (g) (h) 8 (i). Simplify the following epressions, giving your nswer in inde nottion. () (b) (c) (d) (e) (f) (g) (h) (i) (j) 8 (k) 8 (m) (n) (l) 9 9 (o) 0 8. Fill in the missing powers. () 8 (b) (c) (d) (e) 8 (f) (g) (h) (i) 9 (j) (k) (l). Simplify the following, giving your nswers in inde form. () ( ) (b) ( ) (c) ( ) (d) ( ) (e) ( ) (f) ( ) (g) ( ) (h) ( ) (i) ( )
7 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet 8. Fill in the missing numbers. ( ) (b) ( ) (c) 0 ( ) 0 ( ) (e) ( 0 ) 0 (f) ( ) () (d) 9. Simplify ech of the following, giving your nswer in inde nottion. 0 () (b) (c) (d) (e) (f) 8 (g) (h) 8 9 (i) 0 0. Simplify ech of the following epressions. () (b) (c) 0 (d) (e) y y (f) p p (g) q q (h) (i) b b (j) (m) b 0 b (k) c c (l) y (n) y (o) 8 (p) p p p (q) 0 (r) y y y y (s) (t) (u) 8 0 (v) ( ) (w) ( ) () ( ). cn be written s. Find the vlues of p nd q in the following: () p (b) q. Epress s simply s possible: Chllenge! You open book. Two pges fce you. If the product of the two pge numbers is 9, wht re the two pge numbers
8 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B. Fctors A fctor of positive whole number is positive whole number tht will divide ectly into it. Worked Emple List ll the fctors of 0. The fctors of 0 re:,,,, 0, 0 These re ll numbers tht divide ectly into 0. Worked Emple Write the number s the product of two fctors in s mny wys s possible. Eercises. List the fctors of these numbers. () (b) (c) (d) (e) 8 (f) (g) 0 (h) 00 (i) (j) 0 (k) (l) 8. Write ech number below s the product of two fctors in s mny wys s possible. () 0 (b) 8 (c) (d) 9 (e) (f) (g) (h). Fill in the missing numbers. () (b) (c) (d) 0 (e) 0 (f) 88 (g) (h). Here is Bingo crd
9 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet () (b) Circle those numbers tht will divide into ectly. Cross out those numbers tht will divide into ectly () In the row of numbers bove: (i) circle ll numbers divisible by, e.g. 0 (ii) cross out ll numbers divisible by, e.g. (iii) underline ll numbers divisible by. e.g. (b) Describe the numbers which re not circled, crossed out or underlined.. A pttern of counting numbers is shown.,,,, 8, 9, 0,... () (i) Which of these numbers is squre number (ii) Which of these numbers is multiple of nine The pttern is continued. (b) (i) Wht is the net squre number (ii) Wht is the net number tht is multiple of nine B. Prime Fctors, HCF nd LCM Any positive whole number cn be written s the product of number of prime fctors. For emple, 0 Note or 80 A prime number is positive whole number with ectly two fctors; nd itself. The first few prime numbers re,,,,,... Worked Emple Write the number s product of prime numbers. Write s product of two fctors: 8 But 8 9 so 9 But 9 so 8
10 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet This epression contins only prime numbers, so This is clled the product of prime fctors. Another importnt concept is tht of the highest common fctor (HCF) of two (or more) positive integers. The HCF is the lrgest number which is fctor of both (or ll) the numbers. Worked Emple Find the HCF of 0 nd 0. Epressing both 0 nd 0 in terms of their prime fctors gives 0 0 It is esy now to see tht the highest common fctor is. Worked Emple () Write the numbers 0 nd 0 s the product of prime fctors. (b) Find the lrgest common fctor tht will divide into both 0 nd 0. () 0 0 So s product of prime fctors, 0 So s product of prime fctors, 0 0 (b) To find the lrgest common fctor tht will divide into both 0 nd 0, look t the fctors common to ech of the products of primes. The numbers tht pper in both re nd, so the lrgest number tht will divide into both 0 nd 0 is 0. So 0 is the HCF of 0 nd 0. 9
11 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet A relted concept is tht of the lowest common multiple, LCM, of two (or more) positive integers. This is the lowest number into which the two (or ll) numbers cn divide ectly. Worked Emple Find the LCM of nd 0. One wy to find the LCM is to write out multiples of ech number. For emple, multiples of re, 8,, 9, 0,, 8, 9,, 0,... multiples of 0 re 0, 0, 80, 0, 00, 0,... It is esy to see tht 0 is the LCM of nd 0. Another wy is to epress nd 0 in terms of their prime fctors: ( ) ( ) 0 Noting tht ( ) is common to both numbers, the LCM is given by 0 or 0 0. So the LCM 0 Note tht 0 nd 0 0. Eercises. Which of the following re prime numbers,,,,, 9,,, 8, 9,,,. Which numbers between 0 nd 0 re prime numbers. Write ech number below s product of prime fctors. () 0 (b) (c) 8 (d) 8 (e) 0 (f) 0 (g) 9 (h) 8 (i) 00. () Epress nd s the product of prime fctors. (b) By compring the nswers to () find the HCF of nd.. Find the highest common fctors of ech pir of numbers below. (), (b) 0, (c), 0 (d), 0 (e) 0, 80 (f) 0, (g), 0 (h), (i), 0
12 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet. () Epress ech of the following numbers s the product of prime fctors:, 99, (b) (c) By considering the products of the prime fctors, find the highest common fctor of (i) nd 99 (ii) 99 nd (iii) nd Wht is the highest common fctor of ll three numbers. Find the HCF for ech set of three numbers given below. () 0,, 0 (b) 90,, 0 (c),, (d) 00, 0, 0 (e), 8, (f) 0, 0, 0 (g),, (h) 0,, 8 (i) 008, 0, 8. Find the LCM of () nd (b) nd 0 (c) 8 nd 9 (d) 9 nd (e) 0 nd (f) 8 nd 9 9. Find the LCM of ech of the following sets of numbers. () 8,, 0 (b), 8, (c), 0, (d) 9, 8, (e) 90, 80, (f), 0, 8 B. Further Inde Nottion Indices cn lso be negtive or frctions. The rules below eplin how to use these types of indices. n n This is clled the reciprocl of. n n Worked Emple Find: () (b) (c) (d) (e) 8 (d) 9
13 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet () (b) 9 (c) (d) ( ) (e) 8 8 (f) 9 9 Worked Emple Find () (b) m m (d) 8 ( ) (e) b ( ) (c) (f) m + () (b) m m m m m (c) 8 8+ ( ) ( ) (d) 9 ( ) 9
14 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet ( ) (e) b b (f) m ( m ) b m Eercises m. Find s frctions tht do not involve indices, without using clcultor: () (b) (c) (d) (e) 9 (f) (g) (h) (i) (j) (k) (l) (m) 9 (n) (o) 8. Complete the missing numbers, without using clcultor. () (b) (c) 8 (d) (e) (f) 9 (g) (h) (i) (j) (m) (k) (n) m m (l) (o) 00 p 0 p (p) q q (q) q q (r) q q. Use clcultor to find: () 8 (b) 0 (c) (d) (e) (f) 0 (g) 8 (h) (i) (j) (k) 9 (l). Simplify the following epressions, so tht they contin no negtive indices. () (b) (c) 9 (d) (e) ( ) (f) ( )
15 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet ( ) (g) (j) (h) ( ) (k) 9 ( ) ( ) (i) ( ) ( ) (l) (m) b ( ) (n) b (o) b ( ) (p) ( b ) (q) (r) m n b ( ) (s) b 0 (t) m (u) 8 b c (v) m (w) y z () 8 ( b ). () Epress 8 (b) Simplify. s frction in the form, where nd b re integers. b y (c) Find the vlue of y for which. Investigtion Find four integers,, b, c nd d such tht + b + c d.
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