Grade 9 Cambridge – T1 - Lesson 1
Numbers
Natural Numbers ( N ) Real Numbers ( R )
Numbers which are used for counting purpose are called natural numbers.
Eg : 1, 2, 3, 4, 5, 6,……………
Rational Irrational
Whole Numbers
Natural numbers + 0 ( Zero )
Eg : 0 , 1, 2, 3, 4, 5, 6,…………… Fractions Integers ( Z )
Integers ( Z)
Positive natural numbers + negative natural numbers + 0 ( zero )
Negative natural Zero Natural number
Eg : ……..-4, -3 , -2, -1 , 0 , 1, 2, 3, 4, 5, 6,……
number
Numbers are getting larger
Whole number
Rational Numbers
Numbers which are in the form of
𝑝
( 𝑞 ≠ 0 ) 𝑝 & 𝑞 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 /𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 Factors
𝑞
Factors are the numbers you multiply together to get another number.
1 3 5 5 A number can have many factors.
Eg : , , ,− , −2
2 4 1 7 2 × 3 = 6
Irrational numbers
factor Product
Numbers like √2 , 𝜋 ( 𝑃𝑖) which cannot written in fractional form.
• Exact values of an irrational number cannot be written down.
6 factors of 6 -- 1,2,3,6
√1 = 1 Rational 1X6
√2 Irrational 2X3
√3
√4 =2 Rational
√5 Prime Numbers
√6 • Natural numbers that can be divisible only by itself and by (1) one.
Irrational • It has only 2 factors. 1 and number itself.
√7
√8
1- not a prime number
√9 =3 Rational
√10 - Irrational
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Prime factors HCF ( Highest Common factor )/ Greatest Common divisor( GCD)
A factor that is a prime number.
15 Factors of 12 - 1 , 2 , 3 , 4, 6 and 12
1 X 15 Factors of 30 - 1 , 2 , 3 , 5, 6, 10, 15 and 30
3X5
factors of 15 = 1,3,5,15
Prime factors of 15 = 3,5 Common factors Highest Common factor
Product of its prime factors = 3X5
Multiples
Composite Numbers The multiples of a number are obtained by multiplying the number by other
The numbers which have more than 2 factors. natural number.
1 is neither a composite nor prime. Multiples of 12
12 x 1 = 12
Square number 12 x 2 = 24
Is result of multiplying a number by itself. 12 x 3 = 36 …………. And so on
12 - 1
22 - 4
2
3 - 9
42 - 16 Common multiples
2
5 - 25 Multiples that are common to two or more numbers.
Cube number Multiples of 2 - 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18…..
Is a result of multiplying a number by itself two times. Multiples of 3 -3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , 27…..
13 - 1
23 - 8 6 , 12 , 18 …. etc are common multiples
3
3 - 27
43 - 64 Least Common Multiple
LCM
Common factors
Factors that are common to two or more numbers are said to be common factors.
LCM ( least common multiple)
Factors of 12 - 1 , 2 , 3 , 4, 6 and 12 The smallest of the common multiples.
Factors of 30 - 1 , 2 , 3 , 5, 6, 10, 15 and 30
Common factors 1, 2, 3, 6
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Questions
(4024/W/11/15) (580/w/21/15)
3. A number written as the product of its prime factors is 22 × 52 × 7. 7.Work out the highest common factor (HCF) of 36 and 90.
(a) Evaluate this number.
(b) The lowest common multiple of 22 × 52 × 7. and another number, N, is (580/m/22/15)
22 × 3 × 52 × 72 . Find the lowest possible value of N.
2.Find the lowest common multiple (LCM) of 24 and 32.
(4024/S/12/15)
(580/m/12/15)
13 (a) Express 60 as a product of its prime factors.
17. 6, 8 , 14, 36, 47, 50, 130
(b) Find the smallest possible integer m such that 60m is a square number.
From the list of numbers, write down one number for each of the following.
(c) The lowest number that is a multiple of both 60 and the integer n is 180 .
(a) An odd number.
Find the smallest possible value of n.
(b) A square number.
(c) A factor of 70.
(4024/W/12/15)
(d) A multiple of 26.
12 (a) Express 198 as the product of its prime factors.
(b) M = 22 × 3 × 52 N = 23 × 32 × 7
(580/w/32/15)
(i) Find the largest number that divides exactly into M and N.
1 (a) Write down in figures the number twenty one million.
(ii) Find the smallest value of k, such that M × k is a cube number.
(b) Write down the four factors of 21.
(d) Put brackets in this calculation to make it correct.
(580/S/21/15)
(e) Write down the first two prime numbers after 21.
17 (a) Write 30 as a product of its prime factors.
(f) Fill in the missing number.
(b) Find the lowest common multiple (LCM) of 30 and 45. 21 210
210
=
(580/W/42/15)
(h) Work out ( 21)2 .
1. (d) The film costs four million and ninety three thousand dollars to make.
(k) Write down the lowest common multiple (LCM) of 21 and 15.
(i) Write this number in figures.
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(580/s/13/15)
(580/W/31/15) 1. What is the value of the digit 7 in 43 782?
2 (a) Write down a number between 20 and 30 that is 3. Write 0.88 as a fraction in its simplest form.
(i) a multiple of 6, 19. (a) Write down the prime number between 62 and 70.
(ii) a square number, (b) Write 54 as the product of its prime factors.
(iii) a cube number, (c) Find the highest common factor (HCF) of 54 and 90.
(iv) a prime number.
(c) (i) Write 84 as a product of its prime factors. (580/s/32/15)
(ii) Find the highest common factor (HCF) of 84 and 126. 1 (a) 4, 3 , 0 , 2 , 9, 5, 7
From the list above, write down
(580/w/13/15) (i) the factors of 24,
1. Write in figures the number six thousand and fifty four. (ii) a prime factor of 24,
7. 11 , 12 , 13 , 14 , 15, 16 From the list of numbers, write down (iii) the highest common factor (HCF) of 56 and 91,
(a) the factors of 60, (b) (i) Using four numbers from the list in part (a), form the largest 4-digit
(b) the prime numbers. number.
12. Write 72 as a product of its prime factors. (ii) Write your answer to part (b)(i) in words.
(c) Find
(580/m/32/15) (i) the common multiple of 5 and 8 between 100 and 150,
9. (iii) Explain why 6 is not a prime number. (ii) the square number between 350 and 390.
(iv) Explain the term ‘irrational number’.
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